Algebra 2 / Unit 8: Trigonometric Functions
8.3 Trig Values of Any Angle
Notes
- $\displaystyle \tan\theta = \frac{\sin\theta}{\cos\theta}$.
- The reference angle is the acute angle between the terminal side and the $x$-axis.
- Signs by quadrant: I all positive, II sine positive, III tangent positive, IV cosine positive ("All Students Take Calculus").
- Use $\sin^2\theta + \cos^2\theta = 1$ to find one value from another, then pick the sign from the quadrant.
- Example: $\displaystyle \cos\theta = -\frac{5}{13}$ in Quadrant III gives $\displaystyle \sin\theta = -\frac{12}{13}$.
Practice
Evaluate $\displaystyle \tan\frac{3\pi}{4}$.
The point is $\displaystyle \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, so $\displaystyle \tan = \frac{\sin}{\cos} = -1$.
An angle $\theta$ is in Quadrant III. Which is true?
In Quadrant III both $x$ and $y$ are negative, so tangent is positive.
If $\displaystyle \sin\theta = \frac{3}{5}$ and $\theta$ is in Quadrant II, find $\cos\theta$.
$\displaystyle \cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$, and cosine is negative in Quadrant II.
Ready for more? Take the Unit 8 test.