Finding Angles Using Trigonometry

When you know two sides of a right-angled triangle, you can use the inverse trigonometric functions (arcsin, arccos, arctan) to find a missing angle.

Recap: SOH CAH TOA

$\sin\theta = \frac{\text{Opp}}{\text{Hyp}} \qquad \cos\theta = \frac{\text{Adj}}{\text{Hyp}} \qquad \tan\theta = \frac{\text{Opp}}{\text{Adj}}$

To find a missing angle, use the inverse (shift) functions:

$\theta = \sin^{-1}\!\left(\frac{\text{Opp}}{\text{Hyp}}\right) \quad \theta = \cos^{-1}\!\left(\frac{\text{Adj}}{\text{Hyp}}\right) \quad \theta = \tan^{-1}\!\left(\frac{\text{Opp}}{\text{Adj}}\ \right)$

On a calculator, press SHIFT (or 2nd) then sin/cos/tan.

Step-by-Step Method

Label the sides: Hyp (opposite the right angle), Opp (opposite the unknown angle), Adj (next to the unknown angle, not the hyp).
Decide which ratio to use based on which two sides you know.
Calculate the fraction, then press $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ .
Round to the required precision (usually 1 d.p. or nearest degree).

Using tan⁻¹ — opposite and adjacent known

A right triangle has the side opposite the angle = 4 cm and the adjacent side = 3 cm.

We know Opp and Adj → use tan:

$\theta = \tan^{-1}\!\left(\frac{4}{3}\right) = \tan^{-1}(1.333) \approx 53.1°$

Using sin⁻¹ — opposite and hypotenuse known

Opp = 5 cm, Hyp = 13 cm.

$\theta = \sin^{-1}\!\left(\frac{5}{13}\right) \approx \sin^{-1}(0.385) \approx 22.6°$

Warning

Make sure your calculator is set to DEG (degrees), not RAD (radians), unless the question asks for radians.

Real-World Applications

Real world: roof pitch angle

A roofer measures that the roof rises 2 m over 5 m of horizontal run. The pitch angle is:

$\theta = \tan^{-1}\!\left(\dfrac{2}{5}\right) = \tan^{-1}(0.4) \approx 21.8°$

Building codes often specify minimum and maximum pitch angles, so this calculation confirms compliance.

Real world: ski slope gradient

A ski slope drops 30 m vertically over 100 m horizontally. The slope angle is:

$\theta = \tan^{-1}\!\left(\dfrac{30}{100}\right) = \tan^{-1}(0.3) \approx 16.7°$

Piste graders use this to classify runs as green, blue, red, or black based on angle.

Real world: camera angle for a sports photo

A photographer stands 20 m from the base of a 15 m high scoreboard. The angle of elevation to the top is:

$\theta = \tan^{-1}\!\left(\dfrac{15}{20}\right) = \tan^{-1}(0.75) \approx 36.9°$

This tells them how far to tilt the camera upward to frame the scoreboard correctly.

Tip

When to use inverse trig: If you know two sides and want the angle, use $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ — these are the "undo" buttons for the trig functions. Always check your calculator is in degree (DEG) mode.

Practice Problems

A right triangle has Opp = 6 cm and Hyp = 10 cm. Find the angle.

Show answer

θ = sin⁻¹(6/10) = sin⁻¹(0.6) ≈ 36.9°

Adj = 8 m, Hyp = 17 m. Find the angle.

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θ = cos⁻¹(8/17) = cos⁻¹(0.471) ≈ 61.9°

Opp = 7 cm, Adj = 7 cm. What is the angle? (Think before calculating.)

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tan θ = 7/7 = 1, so θ = tan⁻¹(1) = 45°. An isoceles right triangle always has two 45° angles.