Precalculus / Unit 4: Trigonometric Functions
4.3 Right-Triangle Trigonometry
Notes
In a right triangle with acute angle $\theta$: $\displaystyle \sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\displaystyle \cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\displaystyle \tan\theta = \frac{\text{opp}}{\text{adj}}$ (SOH-CAH-TOA).
- Use the Pythagorean theorem to find a missing side.
- If you know $\sin\theta$ and the quadrant, draw a triangle to find $\cos\theta$, remembering the sign.
- Example: in a 5-12-13 triangle, the angle opposite the side 5 has $\displaystyle \sin\theta = \frac{5}{13}$.
Practice
A right triangle has legs 5 and 12 and hypotenuse 13. For the angle opposite the leg of length 5, what is $\cos\theta$?
Adjacent is 12 and the hypotenuse is 13.
A 10 ft ladder leans on a wall and makes a $60^\circ$ angle with the ground. How high up the wall does it reach?
Height $\displaystyle = 10\sin 60^\circ = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}$.
If $\displaystyle \sin\theta = \frac{3}{5}$ and $\theta$ is in quadrant II, what is $\cos\theta$?
The adjacent side is 4, and cosine is negative in quadrant II.
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