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>asin, acos, atan | Common Lisp Nova Spec</TITLE
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</STYLE
></HEAD
><BODY
><DIV
><DIV CLASS="topnav"
><DIV CLASS="breadcrumb"
><SPAN CLASS="breadcrumb-item"
><A HREF="index.html"
>Common Lisp Nova Spec</A
></SPAN
> <SPAN CLASS="breadcrumb-item"
>&#8594; <A HREF="12_Numbers.html"
>12. Numbers</A
></SPAN
> <SPAN CLASS="breadcrumb-item"
>&#8594; <A HREF="12_2_Numbers_Dictionary.html"
>12.2 Numbers Dictionary</A
></SPAN
> <SPAN CLASS="breadcrumb-item"
>&#8594; <A HREF="f_asin.html"
>asin, acos, atan</A
></SPAN
></DIV
><DIV CLASS="apropos"
><DIV CLASS="apropos-io"
><A HREF="f_sin.html" CLASS="prev"
>&#8592;</A
><SPAN ID="apropos-label"
>Apropos </SPAN
><INPUT ID="apropos" AUTOFOCUS="AUTOFOCUS" PLACEHOLDER="Type here to search" ONINPUT="aproposInput(this);" ONKEYUP="aproposKeyup(event);" ONCHANGE="aproposChange(this);" ONFOCUS="aproposFocus(this);" ONFOCUSOUT="aproposFocusout(this);"
><A HREF="v_pi.html" CLASS="next"
>&#8594;</A
></DIV
><DIV ID="apropos-res"
></DIV
></DIV
></DIV
><DIV CLASS="matter"
><DIV CLASS="com"
><DIV CLASS="begincom"
><HR
><TABLE WIDTH="100%" CELLSPACING="0" CELLPADDING="0"
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" WIDTH="100%" CLASS="name"
><SPAN CLASS="idx" DATA-KIND="idxref" DATA-TERM="asin, acos, atan"
></SPAN
><B
>asin, acos, atan</B
></TD
><TD ALIGN="RIGHT" VALIGN="BASELINE" WIDTH="0" NOWRAP="NOWRAP" CLASS="ftype"
><I
>Function</I
></TD
></TR
></TABLE
><HR
></DIV
><UL CLASS="subtoc"
></UL
><DL
><DT
><B
>Syntax</B
></DT
><DD
><P CLASS="j"
><B
>asin</B
> <SPAN CLASS="cmssi"
>number</SPAN
> <SPAN CLASS="arrow"
>&#8594;</SPAN
> <SPAN CLASS="cmssi"
>radians</SPAN
></P
><P CLASS="j"
><B
>acos</B
> <SPAN CLASS="cmssi"
>number</SPAN
> <SPAN CLASS="arrow"
>&#8594;</SPAN
> <SPAN CLASS="cmssi"
>radians</SPAN
></P
><P CLASS="j"
><B
>atan</B
> <SPAN CLASS="cmssi"
>number1</SPAN
> <SPAN CLASS="cmtt"
>&amp;optional</SPAN
> <SPAN CLASS="cmssi"
>number2</SPAN
> <SPAN CLASS="arrow"
>&#8594;</SPAN
> <SPAN CLASS="cmssi"
>radians</SPAN
></P
></DD
><DT
><B
>Arguments and Values</B
></DT
><DD
><P CLASS="j"
><VAR CLASS="param"
>number</VAR
> &#8212; a <A HREF="26_1_Glossary.html#number"
><EM CLASS="term"
>number</EM
></A
>. </P
><P CLASS="j"
><VAR CLASS="param"
>number1</VAR
> &#8212; a <A HREF="26_1_Glossary.html#number"
><EM CLASS="term"
>number</EM
></A
> if <VAR CLASS="param"
>number2</VAR
> is not supplied, or a <EM CLASS="term"
>real</EM
> if <VAR CLASS="param"
>number2</VAR
> is supplied. </P
><P CLASS="j"
><VAR CLASS="param"
>number2</VAR
> &#8212; a <EM CLASS="term"
>real</EM
>. </P
><P CLASS="j"
><VAR CLASS="param"
>radians</VAR
> &#8212; a <A HREF="26_1_Glossary.html#number"
><EM CLASS="term"
>number</EM
></A
> (of radians).</P
></DD
><DT
><B
>Description</B
></DT
><DD
><P CLASS="j"
><A HREF="f_asin.html" CLASS="funref"
><B
>asin</B
></A
>, <A HREF="f_asin.html" CLASS="funref"
><B
>acos</B
></A
>, and <A HREF="f_asin.html" CLASS="funref"
><B
>atan</B
></A
> compute the arc sine, arc cosine, and arc tangent respectively. </P
><P CLASS="j"
>The arc sine, arc cosine, and arc tangent (with only <VAR CLASS="param"
>number1</VAR
> supplied) functions can be defined mathematically for <VAR CLASS="param"
>number</VAR
> or <VAR CLASS="param"
>number1</VAR
> specified as <I CLASS="i"
><I
>x</I
></I
> as in Figure 12&#8211;14.</P
><FIGURE CLASS="boxfig"
><DIV CLASS="figbody"
><TABLE CELLSPACING="0" CELLPADDING="0" RULES="GROUPS" STYLE="margin: 1ex 0" WIDTH="100%"
><THEAD
><TR
><TH ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
><B
>Function</B
></TH
><TH ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <B
>Definition</B
></TH
></TR
></THEAD
><TBODY
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>Arc</SPAN
> <SPAN CLASS="cmr"
>sine</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN STYLE="display:inline-block;position:relative;width:118.045pt;height:17.080pt;vertical-align:-5.400pt;xborder:1px solid #ccc"
><SPAN ID="G24"
>&#161;</SPAN
><SPAN ID="G25"
>i</SPAN
><SPAN ID="G26"
>log</SPAN
><SPAN ID="G27"
>&#161;</SPAN
><SPAN ID="G28"
>ix</SPAN
><SPAN ID="G29"
>+</SPAN
><SPAN ID="G30"
>p</SPAN
><SPAN STYLE="position:absolute;left:79.636pt;top:0.720pt;width:32.908pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G31"
>1</SPAN
><SPAN ID="G32"
>&#161;</SPAN
><SPAN ID="G33"
>x</SPAN
><SPAN ID="G34"
>2</SPAN
><SPAN ID="G35"
>&#162;</SPAN
></SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>Arc</SPAN
> <SPAN CLASS="cmr"
>cosine</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN STYLE="display:inline-block;position:relative;width:88.149pt;height:14.400pt;vertical-align:-4.200pt;xborder:1px solid #ccc"
><SPAN ID="G36"
>(</SPAN
><SPAN ID="G37"
>&#188;</SPAN
><SPAN ID="G38"
>=</SPAN
><SPAN ID="G39"
>2)</SPAN
><SPAN ID="G40"
>&#161;</SPAN
><SPAN ID="G41"
>arcsin</SPAN
><SPAN ID="G42"
>x</SPAN
></SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>Arc</SPAN
> <SPAN CLASS="cmr"
>tangent</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN STYLE="display:inline-block;position:relative;width:160.711pt;height:17.340pt;vertical-align:-5.400pt;xborder:1px solid #ccc"
><SPAN ID="G43"
>&#161;</SPAN
><SPAN ID="G44"
>i</SPAN
><SPAN ID="G45"
>log</SPAN
><SPAN ID="G46"
>&#161;</SPAN
><SPAN ID="G47"
>(1</SPAN
><SPAN ID="G48"
>+</SPAN
><SPAN ID="G49"
>ix</SPAN
><SPAN ID="G50"
>)</SPAN
><SPAN ID="G51"
>p</SPAN
><SPAN STYLE="position:absolute;left:100.970pt;top:0.720pt;width:54.242pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G52"
>1</SPAN
><SPAN ID="G53"
>=</SPAN
><SPAN ID="G54"
>(1</SPAN
><SPAN ID="G55"
>+</SPAN
><SPAN ID="G56"
>x</SPAN
><SPAN ID="G57"
>2</SPAN
><SPAN ID="G58"
>)</SPAN
><SPAN ID="G59"
>&#162;</SPAN
></SPAN
></TD
></TR
></TBODY
></TABLE
></DIV
><FIGCAPTION CLASS="caption"
><B
>Figure 12&#8211;14. Mathematical definition of arc sine, arc cosine, and arc tangent</B
></FIGCAPTION
></FIGURE
><P CLASS="j"
>These formulae are mathematically correct, assuming completely accurate computation. They are not necessarily the simplest ones for real-valued computations. </P
><P CLASS="j"
>If both <VAR CLASS="param"
>number1</VAR
> and <VAR CLASS="param"
>number2</VAR
> are supplied for <A HREF="f_asin.html" CLASS="funref"
><B
>atan</B
></A
>, the result is the arc tangent of <VAR CLASS="param"
>number1</VAR
>/<VAR CLASS="param"
>number2</VAR
>. The value of <A HREF="f_asin.html" CLASS="funref"
><B
>atan</B
></A
> is always between &#8722;&#960; (exclusive) and&#160;&#960; (inclusive) when minus zero is not supported. The range of the two-argument arc tangent when minus zero is supported includes &#8722;&#960;. </P
><P CLASS="j"
>For a <EM CLASS="term"
>real</EM
> <VAR CLASS="param"
>number1</VAR
>, the result is a <EM CLASS="term"
>real</EM
> and lies between &#8722;&#960;<I
>/</I
>2 and&#160;&#960;<I
>/</I
>2 (both exclusive). <VAR CLASS="param"
>number1</VAR
> can be a <A HREF="26_1_Glossary.html#complex"
><EM CLASS="term"
>complex</EM
></A
> if <VAR CLASS="param"
>number2</VAR
> is not supplied. If both are supplied, <VAR CLASS="param"
>number2</VAR
> can be zero provided <VAR CLASS="param"
>number1</VAR
> is not zero. </P
><P CLASS="j"
>The following definition for arc sine determines the range and branch cuts:</P
><CENTER CLASS="displaymath"
><SPAN STYLE="display:inline-block;position:relative;width:182.006pt;height:24.000pt;vertical-align:0.000pt;xborder:1px solid #ccc"
><SPAN ID="G60"
>arcsin</SPAN
><SPAN ID="G61"
>z</SPAN
><SPAN ID="G62"
>=</SPAN
><SPAN ID="G63"
>&#161;</SPAN
><SPAN ID="G64"
>i</SPAN
><SPAN ID="G65"
>log</SPAN
><SPAN ID="G66"
>&#179;</SPAN
><SPAN ID="G67"
>iz</SPAN
><SPAN ID="G68"
>+</SPAN
><SPAN ID="G69"
>p</SPAN
><SPAN STYLE="position:absolute;left:142.681pt;top:2.254pt;width:32.158pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G70"
>1</SPAN
><SPAN ID="G71"
>&#161;</SPAN
><SPAN ID="G72"
>z</SPAN
><SPAN ID="G73"
>2</SPAN
><SPAN ID="G74"
>&#180;</SPAN
></SPAN
></CENTER
><P CLASS="j"
>The branch cut for the arc sine function is in two pieces: one along the negative real axis to the left of&#160;&#8722;1 (inclusive), continuous with quadrant II, and one along the positive real axis to the right of&#160;1 (inclusive), continuous with quadrant IV. The range is that strip of the complex plane containing numbers whose real part is between &#8722;&#960;<I
>/</I
>2 and&#160;&#960;<I
>/</I
>2. A number with real part equal to &#8722;&#960;<I
>/</I
>2 is in the range if and only if its imaginary part is non-negative; a number with real part equal to &#960;<I
>/</I
>2 is in the range if and only if its imaginary part is non-positive. </P
><P CLASS="j"
>The following definition for arc cosine determines the range and branch cuts:</P
><CENTER CLASS="displaymath"
><SPAN STYLE="display:inline-block;position:relative;width:128.545pt;height:23.916pt;vertical-align:0.000pt;xborder:1px solid #ccc"
><SPAN ID="G75"
>arccos</SPAN
><SPAN ID="G76"
>z</SPAN
><SPAN ID="G77"
>=</SPAN
><SPAN ID="G78"
>&#188;</SPAN
><SPAN STYLE="position:absolute;left:61.568pt;top:10.765pt;width:7.271pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G79"
>2</SPAN
><SPAN ID="G80"
>&#161;</SPAN
><SPAN ID="G81"
>arcsin</SPAN
><SPAN ID="G82"
>z</SPAN
></SPAN
></CENTER
><P CLASS="j"
>or, which are equivalent,</P
><CENTER CLASS="displaymath"
><SPAN STYLE="display:inline-block;position:relative;width:186.006pt;height:24.000pt;vertical-align:0.000pt;xborder:1px solid #ccc"
><SPAN ID="G83"
>arccos</SPAN
><SPAN ID="G84"
>z</SPAN
><SPAN ID="G85"
>=</SPAN
><SPAN ID="G86"
>&#161;</SPAN
><SPAN ID="G87"
>i</SPAN
><SPAN ID="G88"
>log</SPAN
><SPAN ID="G89"
>&#179;</SPAN
><SPAN ID="G90"
>z</SPAN
><SPAN ID="G91"
>+</SPAN
><SPAN ID="G92"
>i</SPAN
><SPAN ID="G93"
>p</SPAN
><SPAN STYLE="position:absolute;left:146.681pt;top:2.254pt;width:32.158pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G94"
>1</SPAN
><SPAN ID="G95"
>&#161;</SPAN
><SPAN ID="G96"
>z</SPAN
><SPAN ID="G97"
>2</SPAN
><SPAN ID="G98"
>&#180;</SPAN
></SPAN
></CENTER
><CENTER CLASS="displaymath"
><SPAN STYLE="display:inline-block;position:relative;width:246.595pt;height:30.251pt;vertical-align:0.000pt;xborder:1px solid #ccc"
><SPAN ID="G99"
>arccos</SPAN
><SPAN ID="G100"
>z</SPAN
><SPAN ID="G101"
>=</SPAN
><SPAN ID="G102"
>2</SPAN
><SPAN ID="G103"
>log</SPAN
><SPAN ID="G104"
>&#161;</SPAN
><SPAN ID="G105"
>p</SPAN
><SPAN STYLE="position:absolute;left:110.078pt;top:0.720pt;width:48.108pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G106"
>(1</SPAN
><SPAN ID="G107"
>+</SPAN
><SPAN ID="G108"
>z</SPAN
><SPAN ID="G109"
>)</SPAN
><SPAN ID="G110"
>=</SPAN
><SPAN ID="G111"
>2</SPAN
><SPAN ID="G112"
>+</SPAN
><SPAN ID="G113"
>i</SPAN
><SPAN ID="G114"
>p</SPAN
><SPAN STYLE="position:absolute;left:192.987pt;top:0.720pt;width:48.108pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G115"
>(1</SPAN
><SPAN ID="G116"
>&#161;</SPAN
><SPAN ID="G117"
>z</SPAN
><SPAN ID="G118"
>)</SPAN
><SPAN ID="G119"
>=</SPAN
><SPAN ID="G120"
>2</SPAN
><SPAN ID="G121"
>&#162;</SPAN
><SPAN STYLE="position:absolute;left:61.568pt;top:17.100pt;width:185.027pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G122"
>i</SPAN
></SPAN
></CENTER
><P CLASS="j"
>The branch cut for the arc cosine function is in two pieces: one along the negative real axis to the left of&#160;&#8722;1 (inclusive), continuous with quadrant II, and one along the positive real axis to the right of&#160;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&#160;&#960;. 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 &#960; is in the range if and only if its imaginary part is non-positive. </P
><P CLASS="j"
>The following definition for (one-argument) arc tangent determines the range and branch cuts:</P
><CENTER CLASS="displaymath"
><SPAN STYLE="display:inline-block;position:relative;width:198.739pt;height:27.749pt;vertical-align:0.000pt;xborder:1px solid #ccc"
><SPAN ID="G123"
>arctan</SPAN
><SPAN ID="G124"
>z</SPAN
><SPAN ID="G125"
>=</SPAN
><SPAN ID="G126"
>log</SPAN
><SPAN ID="G127"
>(1</SPAN
><SPAN ID="G128"
>+</SPAN
><SPAN ID="G129"
>iz</SPAN
><SPAN ID="G130"
>)</SPAN
><SPAN ID="G131"
>&#161;</SPAN
><SPAN ID="G132"
>log</SPAN
><SPAN ID="G133"
>(1</SPAN
><SPAN ID="G134"
>&#161;</SPAN
><SPAN ID="G135"
>iz</SPAN
><SPAN ID="G136"
>)</SPAN
><SPAN STYLE="position:absolute;left:61.568pt;top:14.598pt;width:137.171pt;border-bottom:0.960pt solid black;"
></SPAN
><SPAN ID="G137"
>2</SPAN
><SPAN ID="G138"
>i</SPAN
></SPAN
></CENTER
><P CLASS="j"
>Beware of simplifying this formula; &#8220;obvious&#8221; 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 <I
>i</I
> (exclusive), continuous with quadrant II, and one along the negative imaginary axis below &#8722;<I
>i</I
> (exclusive), continuous with quadrant IV. The points <I
>i</I
> and&#160;&#8722;<I
>i</I
> are excluded from the domain. The range is that strip of the complex plane containing numbers whose real part is between &#8722;&#960;<I
>/</I
>2 and&#160;&#960;<I
>/</I
>2. A number with real part equal to &#8722;&#960;<I
>/</I
>2 is in the range if and only if its imaginary part is strictly positive; a number with real part equal to &#960;<I
>/</I
>2 is in the range if and only if its imaginary part is strictly negative. Thus the range of arc tangent is identical to that of arc sine with the points &#8722;&#960;<I
>/</I
>2 and&#160;&#960;<I
>/</I
>2 excluded. </P
><P CLASS="j"
>For <A HREF="f_asin.html" CLASS="funref"
><B
>atan</B
></A
>, the signs of <VAR CLASS="param"
>number1</VAR
> (indicated as <I CLASS="i"
><I
>x</I
></I
>) and <VAR CLASS="param"
>number2</VAR
> (indicated as <I CLASS="i"
><I
>y</I
></I
>) are used to derive quadrant information. Figure 12&#8211;15 details various special cases. The asterisk (*) indicates that the entry in the figure applies to implementations that support minus zero.</P
><FIGURE CLASS="boxfig"
><DIV CLASS="figbody"
><TABLE CELLSPACING="0" CELLPADDING="0" RULES="GROUPS" STYLE="margin: 1ex 0" WIDTH="100%"
><THEAD
><TR
><TH ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
><I CLASS="i"
><I
>y</I
></I
> <B
>Condition</B
></TH
><TH ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I CLASS="i"
><I
>x</I
></I
> <B
>Condition</B
></TH
><TH ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <B
>Cartesian locus</B
></TH
><TH ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <B
>Range of result</B
></TH
></TR
></THEAD
><TBODY
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&gt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Positive</SPAN
> <SPAN CLASS="cmr"
>x-axis</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>0</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>*</SPAN
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>+0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&gt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Positive</SPAN
> <SPAN CLASS="cmr"
>x-axis</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>+0</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>*</SPAN
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&gt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Positive</SPAN
> <SPAN CLASS="cmr"
>x-axis</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmtt"
>&gt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&gt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Quadrant</SPAN
> <SPAN CLASS="cmr"
>I</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>0</SPAN
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>result</SPAN
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>&#960;</SPAN
><I
>/</I
><SPAN CLASS="cmr"
>2</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmtt"
>&gt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Positive</SPAN
> <SPAN CLASS="cmr"
>y-axis</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>&#960;</SPAN
><I
>/</I
><SPAN CLASS="cmr"
>2</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmtt"
>&gt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Quadrant</SPAN
> <SPAN CLASS="cmr"
>II</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>&#960;</SPAN
><I
>/</I
><SPAN CLASS="cmr"
>2</SPAN
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>result</SPAN
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>&#960;</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Negative</SPAN
> <SPAN CLASS="cmr"
>x-axis</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>&#960;</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>*</SPAN
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>+0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Negative</SPAN
> <SPAN CLASS="cmr"
>x-axis</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>+&#960;</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>*</SPAN
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Negative</SPAN
> <SPAN CLASS="cmr"
>x-axis</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>&#960;</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Quadrant</SPAN
> <SPAN CLASS="cmr"
>III</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>&#960;</SPAN
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>result</SPAN
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>&#960;</SPAN
><I
>/</I
><SPAN CLASS="cmr"
>2</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Negative</SPAN
> <SPAN CLASS="cmr"
>y-axis</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>&#960;</SPAN
><I
>/</I
><SPAN CLASS="cmr"
>2</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmtt"
>&gt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Quadrant</SPAN
> <SPAN CLASS="cmr"
>IV</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>&#960;</SPAN
><I
>/</I
><SPAN CLASS="cmr"
>2</SPAN
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>result</SPAN
><SPAN CLASS="cmtt"
>&lt;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Origin</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>undefined</SPAN
> <SPAN CLASS="cmr"
>consequences</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>*</SPAN
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>+0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>+0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Origin</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>+0</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>*</SPAN
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>+0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Origin</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>*</SPAN
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmr"
>+0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Origin</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmr"
>+&#960;</SPAN
></TD
></TR
><TR
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE=";padding-right:3px"
> <SPAN CLASS="cmr"
>*</SPAN
> <I
>y</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <I
>x</I
><SPAN CLASS="cmr"
>=</SPAN
><SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>0</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;padding-right:3px"
> <SPAN CLASS="cmr"
>Origin</SPAN
></TD
><TD ALIGN="LEFT" VALIGN="BASELINE" STYLE="padding-left:3px;"
> <SPAN CLASS="cmsy"
>&#8722;</SPAN
><SPAN CLASS="cmr"
>&#960;</SPAN
></TD
></TR
></TBODY
></TABLE
></DIV
><FIGCAPTION CLASS="caption"
><B
>Figure 12&#8211;15. Quadrant information for arc tangent</B
></FIGCAPTION
></FIGURE
></DD
><DT
><B
>Examples</B
></DT
><DD
><PRE CLASS="screen"
>(asin 0) <SPAN CLASS="cmsy"
><SPAN CLASS="arrow"
>&#8594;</SPAN
></SPAN
> 0.0
(acos #c(0 1)) <SPAN CLASS="cmsy"
><SPAN CLASS="arrow"
>&#8594;</SPAN
></SPAN
> #C(1.5707963267948966 -0.8813735870195432)
(/ (atan 1 (sqrt 3)) 6) <SPAN CLASS="cmsy"
><SPAN CLASS="arrow"
>&#8594;</SPAN
></SPAN
> 0.087266
(atan #c(0 2)) <SPAN CLASS="cmsy"
><SPAN CLASS="arrow"
>&#8594;</SPAN
></SPAN
> #C(-1.5707964 0.54930615)</PRE
></DD
><DT
><B
>Exceptional Situations</B
></DT
><DD
><P CLASS="j"
><A HREF="f_asin.html" CLASS="funref"
><B
>acos</B
></A
> and <A HREF="f_asin.html" CLASS="funref"
><B
>asin</B
></A
> should signal an error of <A HREF="26_1_Glossary.html#type"
><EM CLASS="term"
>type</EM
></A
> <A HREF="t_type-error.html" CLASS="typeref"
><B
>type-error</B
></A
> if <VAR CLASS="param"
>number</VAR
> is not a <A HREF="26_1_Glossary.html#number"
><EM CLASS="term"
>number</EM
></A
>. <A HREF="f_asin.html" CLASS="funref"
><B
>atan</B
></A
> should signal <A HREF="t_type-error.html" CLASS="typeref"
><B
>type-error</B
></A
> if one argument is supplied and that argument is not a <A HREF="26_1_Glossary.html#number"
><EM CLASS="term"
>number</EM
></A
>, or if two arguments are supplied and both of those arguments are not <EM CLASS="term"
>reals</EM
>. </P
><P CLASS="j"
><A HREF="f_asin.html" CLASS="funref"
><B
>acos</B
></A
>, <A HREF="f_asin.html" CLASS="funref"
><B
>asin</B
></A
>, and <A HREF="f_asin.html" CLASS="funref"
><B
>atan</B
></A
> might signal <A HREF="t_arithmetic-error.html" CLASS="typeref"
><B
>arithmetic-error</B
></A
>.</P
></DD
><DT
><B
>See Also</B
></DT
><DD
><P CLASS="j"
><A HREF="f_log.html" CLASS="funref"
><B
>log</B
></A
>, <A HREF="f_sqrt.html" CLASS="funref"
><B
>sqrt</B
></A
>, <A HREF="12_1_Number_Concepts.html#sec_12_1_3_3" CLASS="secref"
><SPAN CLASS="cmr"
>Section</SPAN
> <SPAN CLASS="cmr"
>12.1.3.3</SPAN
> <SPAN CLASS="cmr"
>(Rule</SPAN
> <SPAN CLASS="cmr"
>of</SPAN
> <SPAN CLASS="cmr"
>Float</SPAN
> <SPAN CLASS="cmr"
>Substitutability)</SPAN
></A
></P
></DD
><DT
><B
>Notes</B
></DT
><DD
><P CLASS="j"
>The result of either <A HREF="f_asin.html" CLASS="funref"
><B
>asin</B
></A
> or <A HREF="f_asin.html" CLASS="funref"
><B
>acos</B
></A
> can be a <A HREF="26_1_Glossary.html#complex"
><EM CLASS="term"
>complex</EM
></A
> even if <VAR CLASS="param"
>number</VAR
> is not a <A HREF="26_1_Glossary.html#complex"
><EM CLASS="term"
>complex</EM
></A
>; this occurs when the absolute value of <VAR CLASS="param"
>number</VAR
> is greater than one.</P
></DD
></DL
></DIV
></DIV
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