Trigonometry / Unit 5: Inverse Trigonometric Functions
5.5 Finding Angles with a Calculator
Notes
Use the inverse buttons to find angles from ratios. Check the calculator mode.
- Degree mode gives degrees; radian mode gives radians.
- $\sin^{-1}x$ means the inverse function, not $\displaystyle \frac{1}{\sin x}$. The reciprocal is $\csc x$.
- Example: a 10 ft ramp rising 2 ft has $\sin\theta = 0.2$, so $\theta = \sin^{-1}(0.2) \approx 11.5^\circ$.
- A calculator gives only one angle; other angles with the same ratio need reference angles (see Unit 7).
Practice
A ramp is 10 m long and rises 2 m. What is its angle of inclination to the nearest tenth of a degree?
$\displaystyle \sin\theta = \frac{2}{10} = 0.2$, so $\theta = \sin^{-1}(0.2) \approx 11.5^\circ$.
What does $\sin^{-1}(x)$ mean?
$\sin^{-1}$ is the inverse function, also written arcsin. The reciprocal is cosecant.
A 12 ft ladder has its base 3 ft from a wall. What angle does it make with the ground, to the nearest tenth of a degree?
The base is adjacent to the angle: $\displaystyle \cos\theta = \frac{3}{12} = 0.25$, so $\theta = \cos^{-1}(0.25) \approx 75.5^\circ$.
Ready for more? Take the Unit 5 test.