document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165687>Question 1165687</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Line	    	Quadrant<BR>\n" );
document.write( "y = −x<BR>\n" );
document.write( "    	          II\r<BR>\n" );
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document.write( "sin θ	=	<BR>\n" );
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document.write( "cos θ	=	<BR>\n" );
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document.write( "tan θ	=	<BR>\n" );
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document.write( "csc θ	=	<BR>\n" );
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document.write( "sec θ	=	<BR>\n" );
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document.write( "cot θ	=	<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852944 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852944\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The line $y = -x$ passes through Quadrant II. To find the trigonometric values for an angle $\theta$ whose terminal side lies on this line in Quadrant II, we can choose any point on that segment of the line. \r<BR>\n" );
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document.write( "### 1. Identify a Point $(x, y)$\r<BR>\n" );
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document.write( "The line is $y = -x$. For the line to be in **Quadrant II**, the $x$-coordinate must be **negative** and the $y$-coordinate must be **positive**.\r<BR>\n" );
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document.write( "* Let $x = -1$.<BR>\n" );
document.write( "* Then $y = -(-1) = 1$.<BR>\n" );
document.write( "* The point on the terminal side of the angle is $\mathbf{(-1, 1)}$.\r<BR>\n" );
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document.write( "### 2. Calculate the Radius ($r$)\r<BR>\n" );
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document.write( "The distance from the origin $(0, 0)$ to the point $(x, y)$ is $r$, calculated using the distance formula:<BR>\n" );
document.write( "$$r = \sqrt{x^2 + y^2}$$<BR>\n" );
document.write( "$$r = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}$$\r<BR>\n" );
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document.write( "### 3. Determine the Trigonometric Ratios\r<BR>\n" );
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document.write( "Using the point $(-1, 1)$ and $r=\sqrt{2}$:<BR>\n" );
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document.write( "\sin \theta = \frac{y}{r} \quad \cos \theta = \frac{x}{r} \quad \tan \theta = \frac{y}{x}<BR>\n" );
document.write( "$$\r<BR>\n" );
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document.write( "| Trigonometric Function | Ratio | Value | Rationalized Value |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| **$\sin \theta$** | $y/r$ | $1/\sqrt{2}$ | $\mathbf{\sqrt{2}/2}$ |<BR>\n" );
document.write( "| **$\cos \theta$** | $x/r$ | $-1/\sqrt{2}$ | $\mathbf{-\sqrt{2}/2}$ |<BR>\n" );
document.write( "| **$\tan \theta$** | $y/x$ | $1/(-1)$ | $\mathbf{-1}$ |<BR>\n" );
document.write( "| **$\csc \theta$** | $r/y$ | $\sqrt{2}/1$ | $\mathbf{\sqrt{2}}$ |<BR>\n" );
document.write( "| **$\sec \theta$** | $r/x$ | $\sqrt{2}/(-1)$ | $\mathbf{-\sqrt{2}}$ |<BR>\n" );
document.write( "| **$\cot \theta$** | $x/y$ | $-1/1$ | $\mathbf{-1}$ |\r<BR>\n" );
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document.write( "***<BR>\n" );
document.write( "*Note: The angle $\theta$ represented here is $135^\circ$ or $3\pi/4$ radians.* </SPAN>\n" );
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