Solving Right Triangles
Use trigonometric ratios and inverse trig functions to find all unknown sides and angles of a right triangle. Apply the Pythagorean theorem alongside trig and understand when each tool is most efficient.
A lesson of Geometry, in Right Triangles and Trigonometry, about 10 minutes.
By the end of this lesson, you will be able to find all unknown sides and angles of a right triangle using trigonometric ratios and inverse trig functions, and apply this skill to multi-step geometry problems.
The Big Idea
"Solving" a right triangle means finding all three sides and all three angles. Given any two pieces of information (angle and side, or two sides), you can determine everything else. Trig ratios handle angle-to-side relationships; the Pythagorean theorem and Angle-Sum handle the rest.
The Process
Given an angle $\theta$ and one side:
- Use SOH-CAH-TOA to set up a ratio
- Solve for the unknown side
- Use the Pythagorean theorem or the complementary angle for the remaining pieces
Given two sides:
- Use the Pythagorean theorem for the third side
- Use inverse trig to find an angle
- The other angle is $90° - \theta$ (complementary)
Example 1 — Angle and One Side Given
Right triangle: $\theta = 35°$, hypotenuse $= 20$. Find both legs.
Leg opposite $\theta$:
$$\sin 35° = \frac{\text{opp}}{20} \implies \text{opp} = 20\sin 35° \approx 20(0.5736) \approx 11.47$$
Leg adjacent to $\theta$:
$$\cos 35° = \frac{\text{adj}}{20} \implies \text{adj} = 20\cos 35° \approx 20(0.8192) \approx 16.38$$
Check: $11.47^2 + 16.38^2 \approx 131.6 + 268.3 = 399.9 \approx 400 = 20^2$ ✓
Other angle: $90° - 35° = 55°$
Example 2 — Two Sides Given
Legs: 5 and 12. Solve the triangle completely.
Hypotenuse: $c = \sqrt{25 + 144} = \sqrt{169} = 13$
Angle opposite the leg of length 5:
$$\theta = \tan^{-1}\left(\frac{5}{12}\right) \approx 22.6°$$
Other angle: $90° - 22.6° = 67.4°$
Example 3 — Leg and Angle Given
$\angle A = 52°$, adjacent leg $= 9$. Find the opposite leg and hypotenuse.
Opposite leg:
$$\tan 52° = \frac{\text{opp}}{9} \implies \text{opp} = 9\tan 52° \approx 9(1.2799) \approx 11.52$$
Hypotenuse:
$$\cos 52° = \frac{9}{\text{hyp}} \implies \text{hyp} = \frac{9}{\cos 52°} \approx \frac{9}{0.6157} \approx 14.62$$
Example 4 — Multi-Step
A right triangle has an angle of 40° and the side opposite that angle is 15. Find all other sides and angles.
$$\sin 40° = \frac{15}{\text{hyp}} \implies \text{hyp} = \frac{15}{\sin 40°} \approx \frac{15}{0.6428} \approx 23.34$$
$$\text{adj} = \sqrt{23.34^2 - 15^2} \approx \sqrt{544.8 - 225} \approx \sqrt{319.8} \approx 17.88$$