Homework 7

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. Prove these identities using addition/subtraction/double-angle formulas. Remember: do not assume the statement you are trying to prove. Start from one side or meet in the middle!
    1. $\cos(x + y)\cos(x - y) = \cos^2 x - \sin^2 y$
    2. $\sin(x + y) - \sin(x - y) = 2\cos x \sin y$
    3. $\tan(x - \pi) = \tan x$
      1. Also justify why this statement is true using the graph of tangent.
    4. $1 - \tan x \tan y = \dfrac{\cos(x + y)}{\cos x \cos y}$
    5. $\cos\left(x + \dfrac{\pi}{3}\right) + \sin\left(x - \dfrac{\pi}{6}\right) = 0$
    6. $\sin \left(\dfrac{\pi}{2} - x\right) = \sin \left(\dfrac{\pi}{2} + x\right)$
    7. $(\sin x + \cos x)^2 = 1 + \sin 2x$
    8. $\cos ^4 x - \sin^4 x = \cos 2x$
  2. Use addition/subtraction/double-angle formulas to find the following values. Do not use a calculator.
    1. $\sin 75^\circ$
    2. $\cos \dfrac{11\pi}{12}$
    3. $\sin18^\circ \cos27^\circ + \cos18^\circ\sin27^\circ$
    4. $\cos 195^\circ$
    5. $2 \cos 15^\circ \sin 15^\circ$
    6. $1 - 2\sin^2 22.5^\circ$
    7. $\dfrac{2\tan(-15^\circ)}{1 - \tan^2(-15^\circ)}$
  3. Solve the following equations for $\theta$:
    1. $\cos \theta = \dfrac{\sqrt{3}}{2}$
    2. $\sin \theta = -\dfrac{\sqrt{3}}{2}$
    3. $4 \sin^2 \theta - 1 = 0$
    4. $\sqrt{2}\sin \theta + 1 = 0$
    5. $2\sin^2\theta + 5\sin \theta - 12 = 0$
    6. $2\cos^2\theta - 7\cos\theta + 3 = 0$
    7. $\sqrt{2}\tan\theta\sin\theta - \tan\theta = 0$
    8. (skip) $\cos\dfrac{\theta}{2} - 1 = 0$
    9. (skip) $2\sin\dfrac{\theta}{3} + \sqrt{3} = 0$