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Lesson overview

Geometry — trigonometry

In a right-angled triangle, the ratios of sides relative to an acute angle are constant. For an angle θ: sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent. The acronym SOHCAHTOA helps recall these. To find an unknown side, identify which angle and which sides are known, then select the appropriate ratio. To find an unknown angle, use the inverse trigonometric functions: $\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ . Always identify the hypotenuse first — it is the longest side, opposite the right angle.

Using sine to find the opposite side

In a right-angled triangle, the angle is 35° and the hypotenuse is 10 cm. Find the opposite side.

Identify the angle, the hypotenuse, and which side is opposite. Use $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ .

Using cosine to find the adjacent side

In a right-angled triangle, the angle is 50° and the hypotenuse is 12 m. Find the adjacent side.

Use $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ . The adjacent side touches the angle.

Using tangent to find a side when hypotenuse is unknown

A ladder makes a 60° angle with the ground. The base is 3 m from the wall. Find the height it reaches.

Use $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$ when you don't have (or need) the hypotenuse.

Finding an angle using inverse trigonometry

In a right-angled triangle, the opposite side is 7 cm and the hypotenuse is 12 cm. Find the angle.

Use the inverse function $\sin^{-1}$ , $\cos^{-1}$ or $\tan^{-1}$ to find the angle when you know the ratio.

Worked example — multi-step problem

From the top of a 25 m cliff, a boat is seen at an angle of depression of 12°. How far is the boat from the base of the cliff?

Draw a right-angled triangle with the cliff as the vertical side (25 m) and the distance to the boat as the horizontal side.
The angle of depression from the top equals the angle of elevation from the boat.
The angle is 12°. The opposite side (from the boat's perspective) is the cliff height: 25 m.
Use tangent: $\tan(12°) = \frac{25}{\text{Distance}}$ .
Rearrange: Distance $= \frac{25}{\tan(12°)}$ .
Calculate: Distance $\approx \frac{25}{0.2126} \approx 117.6$ m.
The boat is approximately 117.6 m from the base of the cliff.

Try it

For each problem, identify the right angle, label the angle of interest, and determine which sides are opposite, adjacent and hypotenuse. Choose the appropriate ratio.

Question 1

In a right-angled triangle, the angle is 30° and the hypotenuse is 8 m. Find the opposite side.

💡 Use $\sin(30°) = \frac{\text{Opposite}}{8}$ . Opposite $= 8 \sin(30°) = 8 × 0.5 = 4$ .

Question 2

The adjacent side is 9 cm and the angle is 40°. Find the opposite side.

💡 Use $\tan(40°) = \frac{\text{Opposite}}{9}$ . Opposite $= 9 \tan(40°)$ .

Question 3

In a right-angled triangle, the opposite side is 5 m and the hypotenuse is 13 m. Find the angle.

💡 Use $\sin(θ) = 5/13$ . Then $θ = \sin^{-1}(5/13) ≈ 22.6°$ .

Question 4

A ramp makes a 20° angle with the ground and is 15 m long. Find the vertical height.

💡 The ramp is the hypotenuse. Height $= 15 \sin(20°)$ .

Question 5

In a right-angled triangle, the adjacent side is 7 cm and the hypotenuse is 12 cm. Find the angle.

💡 Use $\cos(θ) = 7/12$ . Then $θ = \cos^{-1}(7/12) ≈ 55.2°$ .

Common mistakes

Watch for these when working through the lesson.

Confusing which side is opposite and which is adjacent. Remember: opposite is away from the angle, adjacent touches the angle.
Using the wrong ratio. Make sure the hypotenuse is involved only for sine and cosine; use tangent when you don't have the hypotenuse.
Forgetting to use the inverse function when finding an angle. Use $\sin^{-1}$ , not sin, to find the angle.
Not identifying the hypotenuse first. It is always the longest side and opposite the right angle.

Trigonometry in right-angled triangles

Lesson overview

Worked example — multi-step problem

Try it

Common mistakes

Related topics

Independent practice will plug in here