Trigonometry in right-angled triangles Cambridge IGCSE Mathematics (9–1) revision
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In plain words
Pythagoras links the sides of a right-angled triangle. Trigonometry links the sides to the angles. With one side and one angle you can find another side; with two sides you can find an angle.
Three things to know
- Label the sides from the angle you're using: the hypotenuse (opposite the right angle), the opposite (across from the angle) and the adjacent (next to it).
- sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent. SOH CAH TOA.
- To find an angle, use the inverse button: sin⁻¹, cos⁻¹ or tan⁻¹. An angle of elevation is measured up from the horizontal; an angle of depression is measured down from it.
Worked example
A right-angled triangle has a hypotenuse of 10 cm. Find the side opposite an angle of 30°.
- You have the hypotenuse and want the opposite, so use sin.
- sin 30° = opposite ÷ 10, so opposite = 10 × sin 30°.
- = 5 cm.
Tips and tricks
- Label the three sides before choosing sin, cos or tan. Pick the one that uses the two sides in the question.
- Make sure your calculator is in degrees.
It lands in your notebook with its questions as flashcards.
Trigonometry in right-angled triangles: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
Which ratio is sin?
SOH: Sine is Opposite over Hypotenuse.
Which ratio is tan?
TOA: Tangent is Opposite over Adjacent.
You know the adjacent side and the hypotenuse. Which ratio finds the angle?
CAH: Cosine is Adjacent over Hypotenuse.
A right-angled triangle has hypotenuse 8 cm. What is the side opposite an angle of 30°?
8 × sin 30° = 8 × 0.5.
What is an angle of elevation measured from?
You look up from the horizontal.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 7 marks. Write your answers on paper, then check them.
Trigonometry in right-angled triangles
Cambridge IGCSE Mathematics (9–1) 0980 · 7 marks · papermunch.org
Name ______________________________ Date ______________
A right-angled triangle has a hypotenuse of 12 cm. Find the side adjacent to an angle of 40°.[2]
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9.19 cm. 12 × cos 40°.
In a right-angled triangle the side opposite angle x is 5 cm and the adjacent side is 8 cm. Find x.[2]
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32.0°. tan⁻¹(5 ÷ 8).
From a point 20 m from the foot of a tree, the angle of elevation of its top is 35°. Find the height of the tree.[3]
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14.0 m. 20 × tan 35°.
Answers: Trigonometry in right-angled triangles
- 1. 9.19 cm. 12 × cos 40°.
- 2. 32.0°. tan⁻¹(5 ÷ 8).
- 3. 14.0 m. 20 × tan 35°.