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. When solving trigonometric equations, what form should you put the equation in first?
  2. 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. $\csc^2\theta - 4 = 0$
      Hint: difference of squares
    9. $4\sin^3\theta=\sin\theta$
  3. (skip this) 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. $\csc^2\theta = \cot \theta + 3$
  4. We can use polar coordinates $r$ and $\theta$ instead of $x$ and $y$. Describe in English what $r$ and $\theta$ measure.
  5. Plot the following polar coordinates:
    1. $(1, 0)$
    2. $(4, \pi / 4)$
    3. $(3, 2\pi/3)$
    4. $(-2, \pi/4)$
    5. $(2, 5\pi/4)$
  6. Find two other polar coordinate representations of $P\left(1, \dfrac{\pi}{4}\right)$
  7. Convert these polar coordinates to rectangular coordinates:
    1. $(6, 2\pi/3)$
    2. $(\sqrt{3}, -5\pi/3)$
    3. $(0, -11\pi)$
  8. Convert these rectangular coordinates to polar:
    1. $(\sqrt{8}, \sqrt{8})$
    2. $(3\sqrt{3}, -3)$
    3. $(0,\pi)$
    4. $(\sqrt{7}, \sqrt{21})$
  9. Convert these functions into polar form and isolate $r$:
    1. $y=x$
    2. $x^2 + y^2 = 9$
    3. $y = 5$
    4. $y = x^2$
  10. Convert these polar equations into rectangular form:
    1. $r = 7$
    2. $r = \dfrac{1}{\sin \theta -\cos \theta}$
    3. $r = 6\cos \theta$
    4. $r = 2 - \cos \theta$
    5. $r^2 = \sin 2\theta$
  11. Sketch a rough graph of the following:
    1. $r = 2$
    2. $\theta = \dfrac{\pi}{2}$
    3. (skip this) $r = 3\sin\theta$
    4. (skip this) $r = 6\cos \theta$
    5. (skip this) $\theta = -\dfrac{\pi}{4}$