Funciones trigonométricas de ángulos

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#003366" TEXT="#F7F7FF"> <div> P r e c á l c u l o 21 | <FONT SIZE="3" COLOR="#00FFCC" FACE="Bell MT" ><A HREF="index.htm" TARGET="_top" TITLE="Precálculo21"><U>Página principal</U></A> </FONT> |  <FONT SIZE="3" COLOR="#00FFCC" FACE="Bell MT" ><A HREF="id418.htm" TARGET="_top" TITLE="_"><U>Contenido</U></A>  </FONT>|  <A HREF="id464.htm" TARGET="_top" TITLE="Apuntes"><U>Apuntes</U></A>  |  <A HREF="id19.htm" TARGET="_top" TITLE=" Trigonometría"><U>Trigonometría </U></A> |</div> </td> </tr><tr> <td valign="top" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="5543838917"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> </td> <td valign="top" height=456 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B> <A NAME="funciones_trigonométricas_de_ángulos"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Funciones trigonométricas de ángulos</B></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:728px;height:90px" data-ad-client="ca-pub-6266400581769473" data-ad-slot="9294650114"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <bR> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="605" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 482 ><NOBR><div> <IMG SRC="fcaf7c80.gif" border=0 width="247" height="200" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <IMG SRC="fe5e1af0.gif" border=0 width="225" height="175" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> ( <I>fig</I>.1)</div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="fcb5ddb0.gif" border=0 width="349" height="475" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="603" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 20 WIDTH="595"><div> <B> <A NAME="signos_de_las_funciones_trigonométricas"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Signos de las funciones trigonométricas</B></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 232 WIDTH="341"><div>           </div> <div>           De acuerdo con el cuadrante en que se halle el lado terminal del ángulo y teniendo en cuenta que la distancia de un punto cualquiera al origen de coordenadas es siempre positiva, y aplicando la "ley de los signos", las funciones trigonométricas pueden ser positivas o negativas. </div> <div>           En la tabla de la parte inferior se resumen los signos de las funciones trigonométricas en cada uno de los cuadrantes.</div> <bR> </tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="fccfae10.gif" border=0 width="250" height="225" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP COLSPAN=2 HEIGHT= 153 ><NOBR><div> <TABLE BORDER="1" CELLPADDING="1" CELLSPACING="0" WIDTH="541" BORDERCOLORLIGHT="#009900" BORDERCOLORDARK="#00FFCC" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 22 WIDTH="73"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="22" ALIGN="bottom" WIDTH="73" HSPACE="0" VSPACE="0"></tD> <tD VALIGN=TOP WIDTH="73"><div> <B>seno</B></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <B>coseno</B></div> </tD> <tD VALIGN=TOP><NOBR><div> <B>tangente</B></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <B>cotangente</B></div> </NOBR></tD> <tD VALIGN=TOP WIDTH="73"><div> <B>secante</B></div> </tD> <tD VALIGN=TOP><NOBR><div> <B>cosecante</B></div> </NOBR></tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> <B>I</B></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> <B>II</B></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fcd0d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fce0d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fcf0d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd00d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> <B>III</B></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd10d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd20d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd30d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd40d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 20 WIDTH="73"><div> <B>IV</B></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd50d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd60d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd70d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> <tD VALIGN=TOP WIDTH="73"><div> +</div> </tD> <tD VALIGN=TOP WIDTH="73"><div> <IMG SRC="fd80d0b0.gif" border=0 width="13" height="11" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </tD> </tR> </TABLE></div> <bR> </NOBR></tD> </tR> </TABLE></div> <div> <B> <A NAME="funciones_trigonométricas_de_un_círculo"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Funciones trigonométricas en un círculo goniométrico:</B></div> <div> Como ya se dijo con  anterioridad, un círculo trigonométrico o goniométrico tiene un radio cuya medida es igual a la unidad. De acuerdo con las deficiones y teniendo en cuenta que la distancia al origen de P es 1, se tiene:</div> <div> <TABLE BORDER="0" CELLPADDING="2" CELLSPACING="1" WIDTH="603" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP HEIGHT= 207 ><NOBR><div> <IMG SRC="ff439b80.gif" border=0 width="313" height="184" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> <tD VALIGN=TOP><NOBR><div> <IMG SRC="ff3fac80.gif" border=0 width="250" height="200" ALT="Imagen de mapa de bits" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> </NOBR></tD> </tR> </TABLE></div> <bR> <div> <TABLE BORDER="2" CELLPADDING="2" CELLSPACING="1" WIDTH="614" BORDERCOLORLIGHT="#C0C0C0" BORDERCOLORDARK="#808080" FRAME="BOX" RULES="ALL" ALIGN="BOTTOM" HSPACE="0" VSPACE="0" > <tR> <tD VALIGN=TOP COLSPAN=3 BGCOLOR=#33CCCC HEIGHT= 101 WIDTH="594"><div> <B> <A NAME="ejercicios_resueltos_1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B>Ejercicios resueltos</B></div> <bR> <div> <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" ><B>          E</B></FONT>n los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >1</FONT> y <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >2</FONT> se dan las coordenadas de <I>P</I>; calcule el valor de las funciones trigonométricas del ángulo correspondiente. En los ejercicios <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >3</FONT> a <FONT SIZE="3" COLOR="#CC0033" FACE="Times New Roman" >6</FONT> deduzca los signos de las funciones trigonométricas para el ángulo que se da.</div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 28 WIDTH="196"><div> <A HREF="#1" TITLE="Funciones trigonométricas de ángulos"><U><IMG SRC="fed71150.gif" border=0 width="113" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <A HREF="#2" TITLE="Funciones trigonométricas de ángulos"><U><IMG SRC="fee74150.gif" border=0 width="116" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <A HREF="#3" TITLE="Funciones trigonométricas de ángulos"><U><IMG SRC="fef55130.gif" border=0 width="85" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> <tR> <tD VALIGN=TOP HEIGHT= 26 WIDTH="196"><div> <A HREF="#4" TITLE="Funciones trigonométricas de ángulos"><U><IMG SRC="ff05d130.gif" border=0 width="93" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <A HREF="#5" TITLE="Funciones trigonométricas de ángulos"><U><IMG SRC="ff15d130.gif" border=0 width="93" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> <tD VALIGN=TOP WIDTH="196"><div> <A HREF="#6" TITLE="Funciones trigonométricas de ángulos"><U><IMG SRC="ff25d130.gif" border=0 width="93" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> </tD> </tR> </TABLE></div> <bR> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B>S o l u c i o n e s</B></div> <bR> <div> <B> <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A></B><B><IMG SRC="fd971150.gif" border=0 width="113" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <div> <B><IMG SRC="fda3cf10.gif" border=0 width="572" height="241" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="fdb74150.gif" border=0 width="116" height="21" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="fdc61030.gif" border=0 width="609" height="259" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="fdd55130.gif" border=0 width="85" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="fde54450.gif" border=0 width="596" height="69" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="fdf5d130.gif" border=0 width="93" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="fe05cd50.gif" border=0 width="604" height="213" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="5"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="fe15d130.gif" border=0 width="93" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="fe255d50.gif" border=0 width="597" height="213" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div>  <A NAME="6"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A><IMG SRC="fe35d130.gif" border=0 width="93" height="19" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div> <IMG SRC="fe459ed0.gif" border=0 width="601" height="237" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <bR> </td> <td valign="top" width=160 BGCOLOR="#FFFFFF" TEXT="#080000"> <bR> <div> <IMG BORDER="0" SRC="Symb075c00b700f000000000.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Funciones trigonométricas de ángulos</div> <div> <IMG BORDER="0" SRC="Symb075c00b700f000000000.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Signos de las funciones trigonométricas</div> <div> <IMG BORDER="0" SRC="Symb075c00b700f000000000.gif" HEIGHT="16" WIDTH="24" ALIGN="bottom" HSPACE="0" VSPACE="0">Funciones trigonométricas en un círculo goniométrico</div> <bR> </td> </tr></table></body>