Homework 7
Directions:
- Show each step of your work and fully simplify each expression.
- 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:
- When solving trigonometric equations, what form should you put the equation in first?
- Solve the following equations for $\theta$:
- $\cos \theta = \dfrac{\sqrt{3}}{2}$
- $\sin \theta = -\dfrac{\sqrt{3}}{2}$
- $4 \sin^2 \theta - 1 = 0$
- $\sqrt{2}\sin \theta + 1 = 0$
- $2\sin^2\theta + 5\sin \theta - 12 = 0$
- $2\cos^2\theta - 7\cos\theta + 3 = 0$
- $\sqrt{2}\tan\theta\sin\theta - \tan\theta = 0$
- $\csc^2\theta - 4 = 0$
Hint: difference of squares
- $4\sin^3\theta=\sin\theta$
- (skip this) Solve the following equations for $\theta$ by using identities:
- $\sin^2\theta = 4 - 2\cos^2\theta$
- $\cos\theta - \sin\theta = 1$
- $\csc^2\theta = \cot \theta + 3$
- We can use polar coordinates $r$ and $\theta$ instead of $x$ and $y$. Describe in English what $r$ and $\theta$ measure.
- Plot the following polar coordinates:
- $(1, 0)$
- $(4, \pi / 4)$
- $(3, 2\pi/3)$
- $(-2, \pi/4)$
- $(2, 5\pi/4)$
- Find two other polar coordinate representations of $P\left(1, \dfrac{\pi}{4}\right)$
- Convert these polar coordinates to rectangular coordinates:
- $(6, 2\pi/3)$
- $(\sqrt{3}, -5\pi/3)$
- $(0, -11\pi)$
- Convert these rectangular coordinates to polar:
- $(\sqrt{8}, \sqrt{8})$
- $(3\sqrt{3}, -3)$
- $(0,\pi)$
- $(\sqrt{7}, \sqrt{21})$
- Convert these functions into polar form and isolate $r$:
- $y=x$
- $x^2 + y^2 = 9$
- $y = 5$
- $y = x^2$
- Convert these polar equations into rectangular form:
- $r = 7$
- $r = \dfrac{1}{\sin \theta -\cos \theta}$
- $r = 6\cos \theta$
- $r = 2 - \cos \theta$
- $r^2 = \sin 2\theta$
- Sketch a rough graph of the following:
- $r = 2$
- $\theta = \dfrac{\pi}{2}$
- (skip this) $r = 3\sin\theta$
- (skip this) $r = 6\cos \theta$
- (skip this) $\theta = -\dfrac{\pi}{4}$