Homework 7

Directions: Turn in your answers in class on a physical piece of paper. Staple multiple sheets together. Feel free to use Desmos for graphing.

Answer the following:

1. Use addition/subtraction formulas to find the following values:
   1. $\sin 75^\circ$
   2. $\cos \dfrac{11\pi}{12}$
   3. $\sin18^\circ \cos27^\circ + \cos18^\circ\sin27^\circ$
2. Prove these identities. Remember: do not assume the statement you are trying to prove. Start from one side!
   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. $\cos\left(x + \dfrac{\pi}{3}\right) + \sin\left(x - \dfrac{\pi}{6}\right) = 0$
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. $\cos\dfrac{\theta}{2} - 1 = 0$
   9. $2\sin\dfrac{\theta}{3} + \sqrt{3} = 0$
4. Solve the following equations for $\theta$ by using identities:
   1. $\sin^2\theta = 4 - 2\cos^2\theta$
   2. $\cos\theta - \sin\theta = 1$
   3. $\csc^2\theta = \cot \theta + 3$
5. Plot the point $(4, \pi / 4)$.
6. Convert these polar coordinates to rectangular coordinates:
   1. $(6, 2\pi/3)$
   2. $(\sqrt{3}, -5\pi/3)$
7. Convert these rectangular coordinates to polar:
   1. $(\sqrt{8}, \sqrt{8})$
   2. $(3\sqrt{3}, -3)$
8. Convert these functions into polar form:
   1. $y=x$
   2. $x^2 + y^2 = 9$
9. Convert these polar equations into rectangular form:
   1. $r = 7$
   2. $r = \dfrac{1}{\sin \theta -\cos \theta}$
   3. $r = 6\cos \theta$