document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163520>Question 1163520</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Point A = (cosθ,sinθ) is at the intersection of x2 +y2 = 1 and a ray starting at the origin that makes an angle, θ, with the positive x-axis. The ray starting at the origin through point P makes an angle of 2θ with the positive x-axis.<BR>\n" );
document.write( "(a) Explain why P = (cos 2θ, sin 2θ).<BR>\n" );
document.write( "(b) Reflect B = (1,0) over the line CA to get an equivalent<BR>\n" );
document.write( "form of the coordinates of P written in terms of cos θ and sin θ.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854376 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</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=854376\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem explores the relationship between trigonometric double-angle identities and geometric transformations.\r<BR>\n" );
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document.write( "### (a) Explaining why $P = (\cos 2\theta, \sin 2\theta)$\r<BR>\n" );
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document.write( "Any point $P$ in the Cartesian plane can be described by its distance from the origin ($r$) and the angle it makes with the positive x-axis ($\alpha$). The coordinates are given by:<BR>\n" );
document.write( "$$P = (r \cos \alpha, r \sin \alpha)$$\r<BR>\n" );
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document.write( "For point $P$:<BR>\n" );
document.write( "1.  **Angle:** The problem states the ray through $P$ makes an angle of **$2\theta$** with the positive x-axis. Thus, $\alpha = 2\theta$.<BR>\n" );
document.write( "2.  **Distance:** While the prompt doesn't explicitly state $P$ is on the unit circle, in the context of deriving trigonometric identities through geometry, $P$ is understood to be the point on the circle $x^2 + y^2 = 1$. Therefore, $r = 1$.\r<BR>\n" );
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document.write( "Substituting these into the coordinate formula gives:<BR>\n" );
document.write( "**$P = (\cos 2\theta, \sin 2\theta)$**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### (b) Reflecting $B=(1,0)$ over the line through the Origin and $A$\r<BR>\n" );
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document.write( "To find the reflection of $B(1,0)$ over the line $OA$ (where $A = (\cos\theta, \sin\theta)$), we observe the geometric symmetry.\r<BR>\n" );
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document.write( "**1. Geometric Reasoning:**<BR>\n" );
document.write( "* The line $OA$ makes an angle of $\theta$ with the positive x-axis.<BR>\n" );
document.write( "* Point $B(1,0)$ is at an angle of $0^\circ$ relative to the x-axis.<BR>\n" );
document.write( "* The angular distance from $B$ to the line of reflection is $\theta - 0 = \theta$.<BR>\n" );
document.write( "* By the definition of reflection, the image $P$ must be the same angular distance on the *other side* of the line.<BR>\n" );
document.write( "* Therefore, the angle of $P$ is $\theta + \theta = 2\theta$.\r<BR>\n" );
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document.write( "**2. Coordinate Calculation via Projection:**<BR>\n" );
document.write( "To reflect a vector $\mathbf{v}$ across a line defined by unit vector $\mathbf{u} = (\cos\theta, \sin\theta)$, we use the formula:<BR>\n" );
document.write( "$$\text{Ref}_{\mathbf{u}}(\mathbf{v}) = 2(\mathbf{v} \cdot \mathbf{u})\mathbf{u} - \mathbf{v}$$\r<BR>\n" );
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document.write( "Here, $\mathbf{v} = (1, 0)$ and $\mathbf{u} = (\cos\theta, \sin\theta)$.<BR>\n" );
document.write( "* **Step 1 (Dot Product):** $\mathbf{v} \cdot \mathbf{u} = (1)(\cos\theta) + (0)(\sin\theta) = \cos\theta$.<BR>\n" );
document.write( "* **Step 2 (Scalar Multiplication):** $2(\mathbf{v} \cdot \mathbf{u})\mathbf{u} = 2\cos\theta(\cos\theta, \sin\theta) = (2\cos^2\theta, 2\sin\theta\cos\theta)$.<BR>\n" );
document.write( "* **Step 3 (Subtraction):** Subtract $\mathbf{v} = (1, 0)$:<BR>\n" );
document.write( "$$P = (2\cos^2\theta - 1, 2\sin\theta\cos\theta)$$\r<BR>\n" );
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document.write( "### Summary of Identities<BR>\n" );
document.write( "By comparing the results of part (a) and part (b), we have derived the double-angle identities:<BR>\n" );
document.write( "* $\cos 2\theta = 2\cos^2\theta - 1$<BR>\n" );
document.write( "* $\sin 2\theta = 2\sin\theta\cos\theta$\r<BR>\n" );
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document.write( "Does the reflection approach make the relationship between these coordinates clearer than just looking at the formulas? </SPAN>\n" );
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