Thread: Trig Problem

(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.

Hello, xwrathbringerx!

Code:

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

(a) Explain why

Draw chord

is inscribed in a semicircle. .Hence: .

In right triangle

Since

(b) Find a similar expression for in terms of

In right triangle

(c) Explain why the coordinates of pt are:

Note that the radius is:

Inscribed is measured by

Central is measured by

Hence: .

In right triangle

Therefore, the coordinates of are: .

(d) Prove that: .

Both are right triangles that have a common acute angle: .

Therefore: .