# Trig Problem

• Nov 20th 2008, 10:55 PM
xwrathbringerx
Trig Problem
http://www.mathhelpforum.com/math-he...rther-trig.bmp

(a) Explain why AP = 2cosθ.

(b) Find a similar expression for BP in terms of θ.

(c) Explain why the coordinates of pt P is (cos2θ, sin2θ).

(d) Prove that triangle APQ is similar to triangle ABP.

(e) Hence deduce that
(i) sin2θ = 2sin θcosθ
(ii) cos2θ = 2cos^2 θ - 1
(iii) tan2θ = (2tanθ)/(1-tan^ 2 θ)

Could someone please help me on this problem?? I've worked out parts a and b but added them in just in case it is required for the other parts.
• Nov 21st 2008, 07:45 PM
Soroban
Hello, xwrathbringerx!

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              * * *           *          *  P         *              o       *          *  * |*               *    *  |       *  * θ    * 2θ  | *     A o - - - - o - - - * o B       *        O      Q *         *                *         *              *           *          *               * * *

(a) Explain why $\displaystyle AP \:=\:2\cos\theta$

Draw chord $\displaystyle PB.$

$\displaystyle \angle APB$ is inscribed in a semicircle. .Hence: .$\displaystyle \angle APB = 90^o$

In right triangle $\displaystyle BPA\!:\;\;\cos\theta \:=\:\frac{AP}{AB}\quad\Rightarrow\quad AP \:=\:AB\!\cdot\cos\theta$

Since $\displaystyle AB = 2\!:\;\;AP \:=\:2\cos\theta$

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(b) Find a similar expression for $\displaystyle BP$ in terms of $\displaystyle \theta.$
In right triangle $\displaystyle BPA\!:\;\;\sin\theta \:=\:\frac{BP}{2} \quad\Rightarrow\quad BP \:=\:2\sin\theta$

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(c) Explain why the coordinates of pt $\displaystyle P$ are: $\displaystyle (\cos2\theta,\:\sin2\theta)$
Note that the radius is: $\displaystyle OP = 1$

Inscribed $\displaystyle \angle PAB = \theta$ is measured by $\displaystyle \tfrac{1}{2}\text{arc}\:\!PB$

Central $\displaystyle \angle POB$ is measured by $\displaystyle \text{arc}\:\!PB$

Hence: .$\displaystyle \angle POB = 2\theta$

In right triangle $\displaystyle PQO\!:\;\begin{array}{c}\cos2\theta = \frac{OQ}{OP} \quad\Rightarrow\quad x \:=\:OQ \:=\:\cos2\theta \\ \\[-4mm] \sin2\theta =\frac{PQ}{OP} \quad\Rightarrow\quad y \:=\:PQ\:=\:\sin2\theta \end{array}$

Therefore, the coordinates of $\displaystyle P$ are: .$\displaystyle (\cos2\theta,\:\sin2\theta)$

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(d) Prove that: .$\displaystyle \Delta APQ\,\sim\,\Delta ABP$
Both are right triangles that have a common acute angle: .$\displaystyle \theta = \angle A$

Therefore: .$\displaystyle \Delta APQ \,\sim\,\Delta ABP$