How to Find an Angle in a Right-Angled Triangle (Using Two Sides)

Find any angle in a right-angled triangle from two known sides: choose the ratio, use the inverse key, and check the size makes sense.

📚 Part of the Trigonometry Clinic: every lesson on trigonometry, elevation, depression and bearings, in one place.

Quick answer: how do you find an angle in a right-angled triangle from two sides?

Name the two sides you know (opposite, adjacent or hypotenuse) from the angle. Choose the ratio that uses those two sides, work out the fraction, then use the inverse key: \theta=\tan^{-1}\!\left(\tfrac{\text{opp}}{\text{adj}}\right), \sin^{-1}\!\left(\tfrac{\text{opp}}{\text{hyp}}\right) or \cos^{-1}\!\left(\tfrac{\text{adj}}{\text{hyp}}\right).

Finding a side is SOHCAHTOA forwards. Finding an angle is SOHCAHTOA backwards: you already know the ratio, and you need the angle that gives it. That backwards step is where many WASSCE candidates slip, writing a ratio such as 0.58 as if it were an angle.

📋 What the WAEC Chief Examiner reported

  • 2018: some candidates treated a trigonometric value as an angle and wrote equations such as θ + 25° = 5.145.
  • 2017: many could not deduce the correct trigonometric ratio from the given diagram.
  • 2023 and 2024: trigonometric ratios were named as a strength, so these marks are within reach.

How to win these marks back

  1. Write the ratio in words first (for example tan θ = opp ÷ adj).
  2. Use the inverse key to turn the ratio into an angle.
  3. Give the angle to the accuracy asked, usually one decimal place.
  4. Check the size: shorter opposite side means a smaller angle.

Source: WAEC Chief Examiners’ Reports, Mathematics (Core), 2017, 2018, 2023, 2024

The five-step method

Five steps to find an angle in a right-angled triangle
The inverse key comes only after the ratio has been chosen and written down.

Example 1: using inverse tan

In a right-angled triangle, the side opposite θ is 5 cm and the side adjacent to θ is 8 cm. Find θ.

Finding an angle from opposite and adjacent using inverse tan
Opposite and adjacent: TOA.
\tan\theta=\dfrac{5}{8}=0.625
\theta=\tan^{-1}(0.625)=32.0^\circ

Example 2: using inverse sin

The side opposite θ is 7 m and the hypotenuse is 12 m. Find θ.

Finding an angle from opposite and hypotenuse using inverse sin
Opposite and hypotenuse: SOH.
\sin\theta=\dfrac{7}{12}=0.5833
\theta=\sin^{-1}(0.5833)=35.7^\circ

Example 3: using inverse cos

The side adjacent to θ is 9 cm and the hypotenuse is 15 cm. Find θ.

Finding an angle from adjacent and hypotenuse using inverse cos
Adjacent and hypotenuse: CAH.
\cos\theta=\dfrac{9}{15}=0.6
\theta=\cos^{-1}(0.6)=53.1^\circ

What sin⁻¹, cos⁻¹ and tan⁻¹ mean

Angle to ratio and ratio to angle: sin and inverse sin on the calculator
The inverse key answers the question “which angle gives this ratio?”

On most scientific calculators you press SHIFT (or 2nd F) and then sin, cos or tan. Make sure the screen shows D or DEG. In radian mode every answer will be wrong.

Check that your angle makes sense

Sense check: compare opposite and adjacent to know if the angle is below or above 45 degrees
Compare the opposite and adjacent sides before you trust the calculator.

The mistake that costs the marks

Common mistake: stopping at the ratio instead of using the inverse key
Stopping at the ratio gives a number, not an angle.
  • Stopping at \sin\theta=0.58 and calling 0.58 the angle.
  • Typing \sin^{-1}(7)\div 12 instead of \sin^{-1}(7\div 12). Use brackets or work out the fraction first.
  • Rounding the fraction too early. Keep four decimal places.
  • Using the wrong pair of sides because θ was not marked first.

Practice questions with answers

  1. The side opposite θ is 4 cm and the side adjacent to θ is 9 cm. Find θ to one decimal place.
  2. The side opposite θ is 11 m and the hypotenuse is 20 m. Find θ.
  3. A ladder 6 m long leans against a wall with its foot 2 m from the wall. Find the angle the ladder makes with the ground.
Show the answers

1. \tan^{-1}(4\div 9) = 24.0°

2. \sin^{-1}(11\div 20) = 33.4°

3. The 2 m is adjacent and the ladder is the hypotenuse: \cos^{-1}(2\div 6) = 70.5°

✏️ Try it yourself

A vertical pole 15 m high casts a shadow 9 m long on level ground. Find the angle of elevation of the sun. Draw the triangle first.

Tap to see the answer

The pole is opposite and the shadow is adjacent, so \theta=\tan^{-1}(15\div 9) = 59.0°.

Frequently asked questions

What is the formula to find an angle of a right triangle?

Use one of \theta=\sin^{-1}\!\left(\tfrac{\text{opp}}{\text{hyp}}\right), \theta=\cos^{-1}\!\left(\tfrac{\text{adj}}{\text{hyp}}\right) or \theta=\tan^{-1}\!\left(\tfrac{\text{opp}}{\text{adj}}\right), depending on which two sides you know.

Can I find the third angle without trigonometry?

Yes. The angles of a triangle add up to 180°, and one angle is 90°, so the other two add up to 90°. Once you have θ, the third angle is 90° − θ.

Why does my calculator give a strange answer?

It is probably in radian mode, or the fraction was not in brackets. Set it to degrees (DEG) and work out the fraction first.

Keep learning

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Dickson Sarpong, SHS Mathematics teacher and founder of The Maths Clinic
Written by Dickson Sarpong

NTC-certified SHS Mathematics teacher in Ghana with WAEC marking experience, and author of The Bridge Series textbooks. Every lesson here is built around the mistakes WAEC examiners keep reporting.

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