About Test 3

Review of Chapters 7-9

Things you should know

  • Relation of unit circle to $\sin \theta$, $\cos \theta$
  • Know how to find other angles with the same $\sin$ or $\cos$
  • Pythagoras' relation for right triangles: $c^2=a^2+b^2$
  • The identity: $$\cos^2(\theta)+\sin^2(\theta)=1$$
  • Know the sides of the $30^o-60^o-90^o$ triangle, and the $45^o-45^o-90^o$ triangle. (And that 30${}^o=\frac{\pi}{6}$ radians; 45${}^o=\frac{\pi}{4}$ radians, etc.)
  • Sum of inside angles = $180^o$ in any triangle
  • Know how ('SOH-CAH-TOA') to find $\cos$, $\sin$, $\tan$ on a right triangle.
  • Sketch a right triangle and use it to solve word problems.
  • This general form of a sinusoidal function: $$f(x)=A \sin\left( B(x-h)\right)+k$$ and how you figure out $k$, $h$, and $A$.
  • How to find the $\sin$ or $\cos$ of very large angles in terms of smaller angles. E.g. to find the $\sin(-25\pi/4)$, you can write the angle as $$\frac{-25 \pi}4=-\frac{-24 \pi }4 -\frac{\pi}4=-6 \pi -\frac{\pi}{4}$$ Since $-6\pi=-3*2\pi$ or 3 full rotations, we'll have... $\sin(-25\pi/4)=\sin(-\pi/4)$, and $\cos(-25\pi/4)=\cos(-\pi/4)$.
  • How to convert radians $\Leftrightarrow$ degrees, using $\pi$ Radians = $180^o$.
  • How to tell if your calculator is in radians or degrees mode!
  • That, $\sin(2.5)$ means $\sin$ of 2.5 Radians. Assume radians if degrees are not explicitly specified.
  • When you use any of the inverse function, $\theta=\arccos x; \theta = \arcsin x;$ etc., be able to figure out what angles (other than what your calculater tells you) are also possible.
  • Recognize graphs of the different trig functions.
  • Go back and forth between: the formula for a sinusoidal $\Leftrightarrow$ graph or sketch.

Relations that I'll make available to you during the test:

 

  • Period, $P$, for that general form of a sinusoidal function and $B$ are related by $$B=\frac{2\pi}{P}$$
  • Angle (in radians) related to radius $r$ and arclength $s$ by: $$\theta = \frac{s}{r}$$
  • Tangent: $\tan \theta=\sin\theta / \cos \theta$
  • $\cot \theta = 1/\tan\theta; \sec \theta = 1/\cos\theta; \csc \theta = 1/\sin\theta$
  • Law of Sines: $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$ (but you should know how the sides and angles are labelled for this to work)
  • Law of Cosines: $$c^2=a^2+b^2-2ab\cos C$$
  • double-angle formulas: $$\sin 2\theta=2\sin\theta\cos\theta$$ $$\cos 2\theta=1-2\sin^2\theta$$