Right-angled trigonometry (SOH CAH TOA)

In a right-angled triangle, sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent (SOH CAH TOA). Label the sides from the angle you are using, pick the ratio that links the two sides involved, then solve.

The key idea

Label the sides relative to the angle θ: the hypotenuse is opposite the right angle, the opposite side is across from θ, and the adjacent side is next to θ.

To find an angle, use the inverse function on your calculator: sin⁻¹, cos⁻¹ or tan⁻¹. Make sure it is set to degrees.

Worked examples

1. Example 1The hypotenuse is 10 cm and θ = 35°. Find the side opposite θ.

   Opposite and hypotenuse: use sine

   1. sin 35° = opp ÷ 10SOH: the two sides involved
   2. opp = 10 × sin 35°Multiply both sides by 10
   3. opp = 10 × 0.5736…Calculator in degrees
   4. opp = 5.74 cmRound to 3 significant figures

   Answer: About 5.74 cm

2. Example 2The opposite side is 7 cm and the adjacent side is 4 cm. Find θ.

   Opposite and adjacent: use tangent

   1. tan θ = 7 ÷ 4TOA
   2. tan θ = 1.75
   3. θ = tan⁻¹(1.75)Inverse tan finds the angle
   4. θ = 60.25…° ≈ 60.3°Round to 1 decimal place

   Answer: About 60.3°

3. Example 3You stand 20 m from a tree and measure the angle of elevation to the top as 40°. How tall is the tree above eye level?

   Angle of elevation: opposite and adjacent

   1. tan 40° = h ÷ 20Height is opposite, distance is adjacent
   2. h = 20 × tan 40°Multiply both sides by 20
   3. h = 20 × 0.8391…
   4. h ≈ 16.8 mAbove eye level. Add your eye height for the full tree.

   Answer: About 16.8 m above eye level

Common mistakes

• Having the calculator in radians instead of degrees.
• Labelling opposite and adjacent from the wrong angle.
• Multiplying when the unknown is on the bottom. If sin 25° = 9 ÷ h, then h = 9 ÷ sin 25°.

Practice questions

1. The hypotenuse is 8 cm and θ = 30°. Find the opposite side.
   Show answer

   4 cm

2. The opposite side is 5 cm and the hypotenuse is 13 cm. Find θ.
   Show answer

   sin⁻¹(5 ÷ 13) ≈ 22.6°

3. θ = 45° and the adjacent side is 6 cm. Find the opposite side.
   Show answer

   6 cm

4. θ = 25° and the opposite side is 9 cm. Find the hypotenuse.
   Show answer

   9 ÷ sin 25° ≈ 21.3 cm

More topic explainers

• Solving linear equations (MYP 2–3)
• Pythagoras’ theorem (MYP 3)
• Solving quadratic equations (MYP 4–5)

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