Inverse Trig and Right Triangles
A ramp rises 3 feet over a 4 foot run, and an inspector needs the angle it makes with the ground. The side ratios are known, but the angle is the unknown. Right triangle trig sends an angle to a ratio. The inverse functions run that machine backward: feed in the ratio, read off the acute angle.
By the end of this page you can:
- Find an unknown acute angle of a right triangle from two known sides.
- Choose the inverse function (arcsin, arccos, or arctan) that matches the sides you have.
- Evaluate a compound expression such as the cosine of an arctangent.
The floor. If you know two sides of a right triangle, the angle is the inverse of the matching ratio: an angle opposite the known opposite over adjacent sides is the arctangent of that ratio.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7e |
| Chapter | 6 |
| Section | 6.4 Inverse Trigonometric Functions and Right Triangles |
| Open alternate | OpenStax Precalculus 2e, 6.3 Inverse Trigonometric Functions |
| Difficulty | Core |
| Time | About 30 minutes |
Before You Start
The Idea
In a right triangle, each acute angle determines its side ratios through sine, cosine, and tangent. Going forward, the angle gives the ratio. To go backward, from a ratio to the angle, apply the inverse function that matches the two sides you have. If you know the opposite and the hypotenuse, use arcsine. Adjacent and hypotenuse, use arccosine. Opposite and adjacent, use arctangent.
Because the angles of a right triangle (other than the right angle) are acute, they fall in the range zero to ninety degrees. That range sits inside the output range of every inverse trig function, so the inverse function returns the angle you want directly. Compound expressions like the cosine of an arctangent are handled the same way: name the inner inverse as an angle, draw or imagine its reference triangle, then read off the requested ratio.
Key Formulas
For an acute angle in a right triangle with sides opposite, adjacent, and hypotenuse measured relative to that angle:
\[ \theta = \sin^{-1}\!\left(\frac{\text{opp}}{\text{hyp}}\right), \quad \theta = \cos^{-1}\!\left(\frac{\text{adj}}{\text{hyp}}\right), \quad \theta = \tan^{-1}\!\left(\frac{\text{opp}}{\text{adj}}\right) \]
Source: Stewart 7e, Section 6.4 Inverse Trigonometric Functions and Right Triangles.
Worked Example
A right triangle has legs of length 3 and 4. Find the acute angle opposite the side of length 3.
- Identify the sides relative to the unknown angle. The side of length 3 is opposite the angle. The other leg, length 4, is adjacent.
- The two known sides are opposite and adjacent, so the matching ratio is tangent: the tangent of the angle equals 3 over 4.
- Apply the inverse: the angle equals the arctangent of 3 over 4.
- Evaluate. 3 divided by 4 is 0.75, and the arctangent of 0.75 is about 36.87 degrees.
Show the answer
The angle is
\[ \theta = \tan^{-1}\!\left(\frac{3}{4}\right) \approx 36.87^\circ. \]
As a check, the other acute angle is the arctangent of 4 over 3, about 53.13 degrees, and the two acute angles sum to 90 degrees as they must in a right triangle.
Connections
This builds on inverse trigonometric functions, which supply the arcsin, arccos, and arctan machinery and their ranges, and on right triangle trigonometry, which is where the opposite, adjacent, and hypotenuse ratios come from. It leads forward to the law of sines, which extends angle and side relationships to triangles that have no right angle. Two common traps to avoid: forgetting to set the calculator to the correct angle mode, and reporting an angle outside the inverse function’s range.
Resources
- Stewart, Redlin, Watson, Precalculus 7e, Section 6.4, primary source for the formulas and worked methods.
- OpenStax Precalculus 2e, Section 6.3 Inverse Trigonometric Functions, open alternate.
Last updated: 2026-06-24