Notes
Any angle
- If $(x, y)$ is on the terminal side and $\displaystyle r = \sqrt{x^2 + y^2}$: $\displaystyle \sin\theta = \frac{y}{r}$, $\displaystyle \cos\theta = \frac{x}{r}$, $\displaystyle \tan\theta = \frac{y}{x}$.
- Signs: Quadrant I all positive, II sine, III tangent, IV cosine (All Students Take Calculus).
The unit circle
- On the unit circle $\cos\theta = x$ and $\sin\theta = y$.
- Key angles: $\displaystyle 30^\circ \to \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, $\displaystyle 45^\circ \to \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, $\displaystyle 60^\circ \to \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.
- To evaluate: use the reference angle, then the sign of the quadrant.
Reciprocals
- $\displaystyle \csc = \frac{1}{\sin}$, $\displaystyle \sec = \frac{1}{\cos}$, $\displaystyle \cot = \frac{1}{\tan}$. They are undefined where the partner is 0.
- Given one function and a quadrant, build a reference triangle for the rest.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle r = \sqrt{64 + 225} = 17$ and $\displaystyle \cos\theta = \frac{x}{r}$.
$\displaystyle \frac{5\pi}{3} = 300^\circ$ is in Quadrant IV with reference angle $60^\circ$, and sine is negative there.
Positive cosine means $x > 0$, and negative tangent then means $y < 0$.
Reference angle $45^\circ$ in Quadrant II, where tangent is negative.
$\cos\pi = -1$, so $\displaystyle \sec\pi = \frac{1}{-1} = -1$.
Positive cosine and negative sine mean Quadrant IV, reference angle $\displaystyle \frac{\pi}{3}$: $\displaystyle 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Consider the angle $\alpha = \frac{7\pi}{6}$ in standard position.
(a) In which quadrant does the terminal side lie, and what is the reference angle?
Show the answer and scoring guide
$\displaystyle \frac{7\pi}{6} = 210^\circ$ is between $180^\circ$ and $270^\circ$, so it is in Quadrant III. The reference angle is $\displaystyle \frac{7\pi}{6} - \pi = \frac{\pi}{6}$.
Check off the points you earned:
(b) Find the exact values of $\sin\alpha$, $\cos\alpha$ and $\tan\alpha$.
Show the answer and scoring guide
In Quadrant III sine and cosine are negative and tangent is positive: $\displaystyle \sin\alpha = -\frac{1}{2}$, $\displaystyle \cos\alpha = -\frac{\sqrt{3}}{2}$, $\displaystyle \tan\alpha = \frac{-1/2}{-\sqrt{3}/2} = \frac{\sqrt{3}}{3}$.
Check off the points you earned:
(c) Find the exact values of $\csc\alpha$, $\sec\alpha$ and $\cot\alpha$.
Show the answer and scoring guide
$\csc\alpha = -2$, $\displaystyle \sec\alpha = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}$, $\displaystyle \cot\alpha = \sqrt{3}$.
Check off the points you earned: