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Introduction</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-2/c-b-character-syntax.html">2. Syntax</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-3/d-b-evaluation.html">3. Evaluation and Compilation</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-4/e-b-introduction.html">4. Types and Classes</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-5/f-b-generalized-reference.html">5. Data and Control Flow</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-6/g-b-the-loop-facility.html">6. Iteration</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-7/h-b-object-creation-and-initialization.html">7. Objects</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../category/81-structures-dictionary.html">8. Structures</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-9/j-b-condition-system-concepts.html">9. Conditions</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-10/ba-b-symbol-concepts.html">10. Symbols</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-11/bb-b-package-concepts.html">11. Packages</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret menu__link--active" aria-expanded="true" href="../bc-b-number-concepts.html">12. 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href="gcd_function.html">gcd</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="incf_decf_macro.html">incf, decf</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="integer-length_function.html">integer-length</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="integer_system-class.html">integer</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="integerp_function.html">integerp</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="lcm_function.html">lcm</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="ldb-test_function.html">ldb-test</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="ldb_accessor.html">ldb</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="log_function.html">log</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="logand_logandc1_logandc2_logeqv_logior_lognand_lognor_lognot_logorc1_logorc2_logxor_function.html">logand, logandc1, logandc2, logeqv, logior, lognand, lognor, lognot, logorc1, logorc2, logxor</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="logbitp_function.html">logbitp</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="logcount_function.html">logcount</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="logtest_function.html">logtest</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="make-random-state_function.html">make-random-state</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="mask-field_accessor.html">mask-field</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="max_min_function.html">max, min</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="minus_function.html">-</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="minusp_plusp_function.html">minusp, plusp</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="mod_rem_function.html">mod, rem</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="most-positive-fixnum_most-negative-fixnum_constant-variable.html">most-positive-fixnum, most-negative-fixnum</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="number_system-class.html">number</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="numberp_function.html">numberp</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="numerator_denominator_function.html">numerator, denominator</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="oneplus_one_function.html">1+, 1</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="parse-integer_function.html">parse-integer</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="phase_function.html">phase</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="plus_function.html">+</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="product_function.html">*</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="random-state-p_function.html">random-state-p</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="random-state_system-class.html">random-state</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="random-state_variable.html">*random-state*</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="random_function.html">random</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="ratio_system-class.html">ratio</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="rational_rationalize_function.html">rational, rationalize</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="rational_system-class.html">rational</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="rationalp_function.html">rationalp</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="real_system-class.html">real</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="realp_function.html">realp</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="realpart_imagpart_function.html">realpart, imagpart</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="short-float-epsilon_short-float-negative-epsilon_single-float-epsilon_single-float-negative-epsilon_double-float-epsilon_double-float-negative-epsilon_long-float-epsilon_long-float-negative-epsilon_con-stant-variable.html">short-float-epsilon, short-float-negative-epsilon, single-float-epsilon, single-float-negative-epsilon, double-float-epsilon, double-float-negative-epsilon, long-float-epsilon, long-float-negative-epsilon</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="short-float_single-float_double-float_long-float_type.html">short-float, single-float, double-float, long-float</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="signed-byte_type.html">signed-byte</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="signum_function.html">signum</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="sin_cos_tan_function.html">sin, cos, tan</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="sinh_cosh_tanh_asinh_acosh_atanh_function.html">sinh, cosh, tanh, asinh, acosh, atanh</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="slash_function.html">/</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="sqrt_isqrt_function.html">sqrt, isqrt</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="unsigned-byte_type.html">unsigned-byte</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="upgraded-complex-part-type_function.html">upgraded-complex-part-type</a></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-3 menu__list-item"><a class="menu__link" tabindex="0" href="zerop_function.html">zerop</a></li></ul></li></ul></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-13/bd-b-character-concepts.html">13. Characters</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-14/be-b-cons-concepts.html">14. Conses</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-15/bf-b-array-concepts.html">15. Arrays</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-16/bg-b-string-concepts.html">16. Strings</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-17/bh-b-sequence-concepts.html">17. Sequences</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-18/bi-b-hash-table-concepts.html">18. Hash Tables</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-19/bj-b-overview-of-filenames.html">19. Filenames</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-20/ca-b-file-system-concepts.html">20. Files</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-21/cb-b-stream-concepts.html">21. Streams</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-22/cc-b-the-lisp-printer.html">22. Printer</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-23/cd-b-reader-concepts.html">23. Reader</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-24/ce-b-system-construction-concepts.html">24. System Construction</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-25/cf-b-the-external-environment.html">25. External Environment</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../environments.html">26. Environments</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../meta-object-protocol.html">27. Meta Object Protocol</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../data-structures.html">28. Data Structures</a></div></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../portability/trivial-packages.html">29. Portability</a></div></li><li class="theme-doc-sidebar-item-link theme-doc-sidebar-item-link-level-1 menu__list-item"><a class="menu__link" href="../../dictionary-entries.html">dictionary-entries</a></li><li class="theme-doc-sidebar-item-category theme-doc-sidebar-item-category-level-1 menu__list-item menu__list-item--collapsed"><div class="menu__list-item-collapsible"><a class="menu__link menu__link--sublist menu__link--sublist-caret" aria-expanded="false" href="../../chap-26/intro.html">Glossary</a></div></li></ul></nav></div></div></aside><main class="docMainContainer_TBSr"><div class="container padding-top--md padding-bottom--lg"><div class="row"><div class="col docItemCol_VOVn"><div class="docItemContainer_Djhp"><article><nav class="theme-doc-breadcrumbs breadcrumbsContainer_Z_bl" aria-label="Breadcrumbs"><ul class="breadcrumbs" itemscope="" itemtype="https://schema.org/BreadcrumbList"><li class="breadcrumbs__item"><a aria-label="Home page" class="breadcrumbs__link" href="../../index.html"><svg viewBox="0 0 24 24" class="breadcrumbHomeIcon_YNFT"><path d="M10 19v-5h4v5c0 .55.45 1 1 1h3c.55 0 1-.45 1-1v-7h1.7c.46 0 .68-.57.33-.87L12.67 3.6c-.38-.34-.96-.34-1.34 0l-8.36 7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 1-1z" fill="currentColor"></path></svg></a></li><li class="breadcrumbs__item"><span class="breadcrumbs__link">12. Numbers</span><meta itemprop="position" content="1"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item"><a class="breadcrumbs__link" itemprop="item" href="../../category/122-numbers-dictionary.html"><span itemprop="name">12.2 Numbers Dictionary</span></a><meta itemprop="position" content="2"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link" itemprop="name">asin, acos, atan</span><meta itemprop="position" content="3"></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><h1>asin, acos, atan</h1>
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<p><strong>asin, acos, atan</strong> <span><i>Function</i></span></p>
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<p><strong>Syntax:</strong></p>
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<p><span><b>asin</b></span> <em>number → radians</em></p>
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<p><span><b>acos</b></span> <em>number → radians</em></p>
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<p><span><b>atan</b></span> <em>number1</em> &optional <em>number2 → radians</em></p>
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<p><strong>Arguments and Values:</strong></p>
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<p><span><i>number</i></span>—a <span><i>number</i></span> .</p>
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<p><em>number1</em>—a <span><i>number</i></span> if <em>number2</em> is not supplied, or a <em>real</em> if <em>number2</em> is supplied.</p>
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<p><em>number2</em>—a <em>real</em>.</p>
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<p><em>radians</em>—a <span><i>number</i></span> (of radians).</p>
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<p><strong>Description:</strong></p>
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<p><span><b>asin</b></span>, <span><b>acos</b></span>, and <span><b>atan</b></span> compute the arc sine, arc cosine, and arc tangent respectively.</p>
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<p>The arc sine, arc cosine, and arc tangent (with only <em>number1</em> supplied) functions can be defined mathematically for <span><i>number</i></span> or <em>number1</em> specified as <em>x</em> as in Figure 12–14.</p>
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<p>|<b><sup>Function Definition</sup></b><i>√</i> |</p>
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<p>| :- |</p>
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<p>|</p><p>Arc sine <em>−i</em> log <em>ix</em> +1 <em>− x</em><sup>2</sup> </p><p>Arc cosine (*π/*2) <em>−</em> arcsin <em>x</em> </p><p><sub>Arc tangent <i>−i</i> log (1 + <i>ix</i>)</sub><sup>p</sup>1<i>/</i>(1 + <i>x</i><sup>2</sup><sub>)</sub> </p>|<p></p>
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<p><strong>Figure 12–14. Mathematical definition of arc sine, arc cosine, and arc tangent</strong></p>
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<p>These formulae are mathematically correct, assuming completely accurate computation. They are not necessarily the simplest ones for real-valued computations.</p>
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<p>If both <em>number1</em> and <em>number2</em> are supplied for <span><b>atan</b></span>, the result is the arc tangent of <em>number1</em>/<em>number2</em>. The value of <span><b>atan</b></span> is always between <em>−π</em> (exclusive) and <em>π</em> (inclusive) when minus zero is not supported. The range of the two-argument arc tangent when minus zero is supported includes <em>−π</em>.</p>
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<p>For a <em>real number1</em>, the result is a <em>real</em> and lies between *−π/*2 and *π/*2 (both exclusive). <em>number1</em> can be a <span><i>complex</i></span> if <em>number2</em> is not supplied. If both are supplied, <em>number2</em> can be zero provided <em>number1</em> is not zero.</p>
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<p>The following definition for arc sine determines the range and branch cuts:</p>
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<span><b>asin, acos, atan</b></span>
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<p>arcsin <em>z</em> = <em>−i</em> log</p>
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<p><sub><i>iz</i> +</sub><sup>p</sup>1 <i>− z</i><sup>2</sup></p>
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<p>The branch cut for the arc sine function is in two pieces: one along the negative real axis to the left of <em>−</em>1 (inclusive), continuous with quadrant II, and one along the positive real axis to the right of 1 (inclusive), continuous with quadrant IV. The range is that strip of the complex plane containing numbers whose real part is between *−π/*2 and *π/*2. A number with real part equal to *−π/*2 is in the range if and only if its imaginary part is non-negative; a number with real part equal to *π/*2 is in the range if and only if its imaginary part is non-positive.</p>
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<p>The following definition for arc cosine determines the range and branch cuts: <sub>arccos <i>z</i> =</sub><i><sup>π</sup></i><sub>2</sub><i>−</i> arcsin <i>z</i></p>
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<p>or, which are equivalent,</p>
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<p>arccos <em>z</em> = <em>−i</em> log</p>
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<p><sub><i>z</i> + <i>i</i></sub><sup>p</sup>1 <i>− z</i><sup>2</sup></p>
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<sub>arccos <i>z</i> =2 log</sub>
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<sup>p</sup>
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<sub>(1 + <i>z</i>)<i>/</i>2 + <i>i</i></sub>
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<sup>p</sup>
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<sub>(1 <i>− z</i>)<i>/</i>2</sub>
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<p><em>i</em></p>
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<p>The branch cut for the arc cosine function is in two pieces: one along the negative real axis to the left of <em>−</em>1 (inclusive), continuous with quadrant II, and one along the positive real axis to the right of 1 (inclusive), continuous with quadrant IV. This is the same branch cut as for arc sine. The range is that strip of the complex plane containing numbers whose real part is between 0 and <em>π</em>. A number with real part equal to 0 is in the range if and only if its imaginary part is non-negative; a number with real part equal to <em>π</em> is in the range if and only if its imaginary part is non-positive.</p>
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<p>The following definition for (one-argument) arc tangent determines the range and branch cuts:</p>
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||
<p>arctan <i>z</i> =<sup>log (1 + <i>iz</i>) <i>−</i> log (1 <i>− iz</i>)</sup></p>
|
||
<p>2<em>i</em></p>
|
||
<p>Beware of simplifying this formula; “obvious” simplifications are likely to alter the branch cuts or the values on the branch cuts incorrectly. The branch cut for the arc tangent function is in two pieces: one along the positive imaginary axis above <em>i</em> (exclusive), continuous with quadrant</p>
|
||
<p>II, and one along the negative imaginary axis below <em>−i</em> (exclusive), continuous with quadrant IV. The points <em>i</em> and <em>−i</em> are excluded from the domain. The range is that strip of the complex plane containing numbers whose real part is between *−π/*2 and *π/*2. A number with real part equal to *−π/*2 is in the range if and only if its imaginary part is strictly positive; a number with real part equal to *π/*2 is in the range if and only if its imaginary part is strictly negative. Thus the range of</p>
|
||
<p>arc tangent is identical to that of arc sine with the points *−π/*2 and *π/*2 excluded.</p>
|
||
<span><b>asin, acos, atan</b></span>
|
||
<p>For <span><b>atan</b></span>, the signs of <em>number1</em> (indicated as <em>x</em>) and <em>number2</em> (indicated as <em>y</em>) are used to derive quadrant information. Figure 12–15 details various special cases. The asterisk (*) indicates that the entry in the figure applies to implementations that support minus zero.</p>
|
||
<p>|<em>y</em> <strong>Condition</strong> <em>x</em> <strong>Condition Cartesian locus Range of result</strong>|</p>
|
||
<p>| :- |</p>
|
||
<p>|</p><p><em>y</em> = 0 <em>x</em> > 0 Positive x-axis 0 </p><p>* <em>y</em> = +0 <em>x</em> > 0 Positive x-axis +0 </p><p>* <em>y</em> = <em>−</em>0 <em>x</em> > 0 Positive x-axis <em>−</em>0 </p><p><em>y</em> > 0 <em>x</em> > 0 Quadrant I 0 < result < *π/*2 <em>y</em> > 0 <em>x</em> = 0 Positive y-axis *π/*2 </p><p><em>y</em> > 0 <em>x</em> < 0 Quadrant II *π/*2 < result < <em>π y</em> = 0 <em>x</em> < 0 Negative x-axis <em>π</em> </p><p>* <em>y</em> = +0 <em>x</em> < 0 Negative x-axis +<em>π</em> </p><p>* <em>y</em> = <em>−</em>0 <em>x</em> < 0 Negative x-axis <em>−π</em> </p><p><em>y</em> < 0 <em>x</em> < 0 Quadrant III <em>−π</em> < result < *−π/*2 <em>y</em> < 0 <em>x</em> = 0 Negative y-axis *−π/*2 </p><p><em>y</em> < 0 <em>x</em> > 0 Quadrant IV *−π/*2 < result < 0 <em>y</em> = 0 <em>x</em> = 0 Origin undefined consequences * <em>y</em> = +0 <em>x</em> = +0 Origin +0 </p><p>* <em>y</em> = <em>−</em>0 <em>x</em> = +0 Origin <em>−</em>0 </p><p>* <em>y</em> = +0 <em>x</em> = <em>−</em>0 Origin +<em>π</em> </p><p>* <em>y</em> = <em>−</em>0 <em>x</em> = <em>−</em>0 Origin <em>−π</em></p>|<p></p>
|
||
<p><strong>Figure 12–15. Quadrant information for arc tangent</strong></p>
|
||
<p><strong>Examples:</strong></p>
|
||
<div class="language-lisp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-lisp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token plain"> </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">asin</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> → </span><span class="token number" style="color:#36acaa">0.0</span><span class="token plain"> </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">acos</span><span class="token plain"> #c</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">0</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">1</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> → #C</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">1.5707963267948966</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">-0.8813735870195432</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">/</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">atan</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">1</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">sqrt</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">3</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">6</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> → </span><span class="token number" style="color:#36acaa">0.087266</span><span class="token plain"> </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">atan</span><span class="token plain"> #c</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">0</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">2</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> → #C</span><span class="token punctuation" style="color:#393A34">(</span><span class="token number" style="color:#36acaa">-1.5707964</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">0.54930615</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p><strong>Exceptional Situations:</strong></p>
|
||
<p><span><b>acos</b></span> and <span><b>asin</b></span> should signal an error of <span><i>type</i></span> <span><b>type-error</b></span> if <span><i>number</i></span> is not a <span><i>number</i></span> . <span><b>atan</b></span> should signal <span><b>type-error</b></span> if one argument is supplied and that argument is not a <span><i>number</i></span> , or if two arguments are supplied and both of those arguments are not <em>reals</em>.</p>
|
||
<p><span><b>acos</b></span>, <span><b>asin</b></span>, and <span><b>atan</b></span> might signal <span><b>arithmetic-error</b></span>.</p>
|
||
<p><strong>See Also:</strong></p>
|
||
<p><span><b>log</b></span>, <span><b>sqrt</b></span>, Section 12.1.3.3 (Rule of Float Substitutability)</p>
|
||
<p><strong>Notes:</strong></p>
|
||
<p>The result of either <span><b>asin</b></span> or <span><b>acos</b></span> can be a <span><i>complex</i></span> even if <span><i>number</i></span> is not a <span><i>complex</i></span> ; this occurs when the absolute value of <span><i>number</i></span> is greater than one.</p>
|
||
<p><strong>pi</strong> <span><i>Constant Variable</i></span></p>
|
||
<p><strong>Value:</strong></p>
|
||
<p>an <em>implementation-dependent long float</em>.</p>
|
||
<p><strong>Description:</strong></p>
|
||
<p>The best <span><i>long float</i></span> approximation to the mathematical constant <em>π</em>.</p>
|
||
<p><strong>Examples:</strong></p>
|
||
<div class="language-lisp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-lisp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token comment" style="color:#999988;font-style:italic">;; In each of the following computations, the precision depends </span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token comment" style="color:#999988;font-style:italic">;; on the implementation. Also, if ‘long float’ is treated by </span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token comment" style="color:#999988;font-style:italic">;; the implementation as equivalent to some other float format </span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token comment" style="color:#999988;font-style:italic">;; (e.g., ‘double float’) the exponent marker might be the marker </span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token comment" style="color:#999988;font-style:italic">;; for that equivalent (e.g., ‘D’ instead of ‘L’). </span><span class="token plain"></span><br></span><span class="token-line" style="color:#393A34"><span class="token plain">pi → 3.141592653589793L0 </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">cos</span><span class="token plain"> pi</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> → -1.0L0 </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"></span><span class="token punctuation" style="color:#393A34">(</span><span class="token defun keyword" style="color:#00009f">defun</span><span class="token defun"> </span><span class="token defun function" style="color:#d73a49">sin-of-degrees</span><span class="token defun"> </span><span class="token defun punctuation" style="color:#393A34">(</span><span class="token defun arguments argument variable" style="color:#36acaa">degrees</span><span class="token defun punctuation" style="color:#393A34">)</span><span class="token plain"> </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">let</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">x</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token keyword" style="color:#00009f">if</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">floatp</span><span class="token plain"> degrees</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> degrees </span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">float</span><span class="token plain"> degrees pi</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><br></span><span class="token-line" style="color:#393A34"><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">sin</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token plain">\* x </span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">/</span><span class="token plain"> </span><span class="token punctuation" style="color:#393A34">(</span><span class="token car">float</span><span class="token plain"> pi x</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><span class="token number" style="color:#36acaa">180</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token punctuation" style="color:#393A34">)</span><span class="token plain"> </span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg viewBox="0 0 24 24" class="copyButtonSuccessIcon_LjdS"><path fill="currentColor" d="M21,7L9,19L3.5,13.5L4.91,12.09L9,16.17L19.59,5.59L21,7Z"></path></svg></span></button></div></div></div>
|
||
<p><strong>Notes:</strong></p>
|
||
<p>An approximation to <em>π</em> in some other precision can be obtained by writing (float pi x), where x is a <span><i>float</i></span> of the desired precision, or by writing (coerce pi <span><i>type</i></span>), where <span><i>type</i></span> is the desired type, such as <span><b>short-float</b></span>.</p>
|
||
<p><strong>sinh, cosh, tanh, asinh, acosh, atanh</strong></p>
|
||
<h2 class="anchor anchorWithStickyNavbar_LWe7" id="expanded-reference-asin-acos-atan">Expanded Reference: asin, acos, atan<a href="asin_acos_atan_function.html#expanded-reference-asin-acos-atan" class="hash-link" aria-label="Direct link to Expanded Reference: asin, acos, atan" title="Direct link to Expanded Reference: asin, acos, atan"></a></h2>
|
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<div class="theme-admonition theme-admonition-tip admonition_xJq3 alert alert--success"><div class="admonitionHeading_Gvgb"><span class="admonitionIcon_Rf37"><svg viewBox="0 0 12 16"><path fill-rule="evenodd" d="M6.5 0C3.48 0 1 2.19 1 5c0 .92.55 2.25 1 3 1.34 2.25 1.78 2.78 2 4v1h5v-1c.22-1.22.66-1.75 2-4 .45-.75 1-2.08 1-3 0-2.81-2.48-5-5.5-5zm3.64 7.48c-.25.44-.47.8-.67 1.11-.86 1.41-1.25 2.06-1.45 3.23-.02.05-.02.11-.02.17H5c0-.06 0-.13-.02-.17-.2-1.17-.59-1.83-1.45-3.23-.2-.31-.42-.67-.67-1.11C2.44 6.78 2 5.65 2 5c0-2.2 2.02-4 4.5-4 1.22 0 2.36.42 3.22 1.19C10.55 2.94 11 3.94 11 5c0 .66-.44 1.78-.86 2.48zM4 14h5c-.23 1.14-1.3 2-2.5 2s-2.27-.86-2.5-2z"></path></svg></span>tip</div><div class="admonitionContent_BuS1"><p>TODO: Please contribute to this page by adding explanations and examples</p></div></div>
|
||
<div class="language-lisp codeBlockContainer_Ckt0 theme-code-block" style="--prism-color:#393A34;--prism-background-color:#f6f8fa"><div class="codeBlockContent_biex"><pre tabindex="0" class="prism-code language-lisp codeBlock_bY9V thin-scrollbar" style="color:#393A34;background-color:#f6f8fa"><code class="codeBlockLines_e6Vv"><span class="token-line" style="color:#393A34"><span class="token punctuation" style="color:#393A34">(</span><span class="token car">asin</span><span class="token plain">, acos, atan </span><span class="token punctuation" style="color:#393A34">)</span><br></span></code></pre><div class="buttonGroup__atx"><button type="button" aria-label="Copy code to clipboard" title="Copy" class="clean-btn"><span class="copyButtonIcons_eSgA" aria-hidden="true"><svg viewBox="0 0 24 24" class="copyButtonIcon_y97N"><path fill="currentColor" d="M19,21H8V7H19M19,5H8A2,2 0 0,0 6,7V21A2,2 0 0,0 8,23H19A2,2 0 0,0 21,21V7A2,2 0 0,0 19,5M16,1H4A2,2 0 0,0 2,3V17H4V3H16V1Z"></path></svg><svg 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