Trigonometry / Unit 3: The Unit Circle and Trig Functions of Any Angle
3.6 Finding All Six Functions
Notes
Given one function and the quadrant, sketch a reference triangle to get the others.
- Choose $x$, $y$, $r$ that fit the ratio, find the missing one with $x^2 + y^2 = r^2$, then attach signs from the quadrant.
- Example: $\displaystyle \cos\theta = \frac{3}{5}$ in Quadrant IV gives $x = 3$, $r = 5$, $y = -4$, so $\displaystyle \sin\theta = -\frac{4}{5}$.
- If the quadrant is not given, use the signs of other functions to decide.
Practice
Suppose $\displaystyle \sin\theta = \frac{3}{5}$ and $\theta$ is in Quadrant II. What is $\cos\theta$?
$y = 3$, $r = 5$, so $|x| = 4$, and $x < 0$ in Quadrant II.
Suppose $\displaystyle \tan\theta = \frac{5}{12}$ and $\theta$ is in Quadrant III. What is $\sin\theta$?
In Quadrant III, $x = -12$, $y = -5$, $r = 13$, so $\displaystyle \sin\theta = -\frac{5}{13}$.
If $\displaystyle \cos\theta = -\frac{1}{2}$ and $\sin\theta > 0$, what is $\theta$ in $[0, 2\pi)$?
Cosine negative and sine positive means Quadrant II, with reference angle $\displaystyle \frac{\pi}{3}$: $\displaystyle \pi - \frac{\pi}{3} = \frac{2\pi}{3}$.
Ready for more? Take the Unit 3 test.