Solving trigonometric equations Cambridge IGCSE Additional Mathematics revision
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In plain words
sin x = 0.5 does not have one answer. It has two in every full turn, and the calculator only gives you the first. The skill is finding the rest, and only the ones inside the range you are given.
6 things to know
- The calculator gives the principal value. The second solution comes from the symmetry of the graph.
- For sin: the solutions are θ and 180° − θ. For cos: θ and 360° − θ. For tan: θ and θ + 180°.
- Add or subtract 360° to find any further solutions in the range.
- For an equation in 2x or 3x, change the range first. If 0° ≤ x ≤ 180°, then 0° ≤ 2x ≤ 360°. Find every value of 2x in that range, then halve them.
- If the equation mixes functions, use an identity to get one function only. A quadratic in sin x, cos x or tan x is then factorised.
- Change sec, cosec and cot into cos, sin and tan before solving.
Worked example
Solve 2cos²x − cos x − 1 = 0 for 0° ≤ x ≤ 360°.
- Factorise: (2cos x + 1)(cos x − 1) = 0.
- cos x = −1/2 gives x = 120° and x = 240°.
- cos x = 1 gives x = 0° and x = 360°.
- x = 0°, 120°, 240° or 360°.
Worked example
Solve 4 cot θ = tan θ for 0° ≤ θ ≤ 360°.
- Write cot θ as 1 ÷ tan θ: 4 ÷ tan θ = tan θ, so tan²θ = 4.
- tan θ = 2 gives θ = 63.4° and 243.4°.
- tan θ = −2 gives θ = 116.6° and 296.6°.
Tips and tricks
- Never divide both sides by sin x or cos x: solutions are lost. Factorise instead.
- Taking a square root gives two values, positive and negative. Both give solutions.
It lands in your notebook with its questions as flashcards.
Solving trigonometric equations: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What are the solutions of sin x = 0.5 for 0° ≤ x ≤ 360°?
For sine, the second solution is 180° − θ.
cos x = 0.5 has the solution 60°. What is the other solution between 0° and 360°?
For cosine, the second solution is 360° − θ.
tan x = 1 has the solution 45°. What is the other solution between 0° and 360°?
For tangent, add 180°.
To solve sin 2x = 0.5 for 0° ≤ x ≤ 180°, what range is needed for 2x?
Double both ends of the range.
How should cosec x = 2 be rewritten before solving?
cosec x = 1 ÷ sin x.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
4 questions, 13 marks. Write your answers on paper, then check them.
Solving trigonometric equations
Cambridge IGCSE Additional Mathematics 0606 · 13 marks · papermunch.org
Name ______________________________ Date ______________
Solve cos x = −0.5 for 0° ≤ x ≤ 360°.[3]
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x = 120° or x = 240°.
Solve tan 2x = 1 for 0° ≤ x ≤ 180°.[3]
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x = 22.5° or x = 112.5°. 2x = 45° or 225°.
Solve 2 sin²x + sin x − 1 = 0 for 0° ≤ x ≤ 360°.[4]
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x = 30°, 150° or 270°. (2 sin x − 1)(sin x + 1) = 0, so sin x = 1/2 or sin x = −1.
Solve sec x = 2 for 0° ≤ x ≤ 360°.[3]
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x = 60° or x = 300°. cos x = 1/2.
Answers: Solving trigonometric equations
- 1. x = 120° or x = 240°.
- 2. x = 22.5° or x = 112.5°. 2x = 45° or 225°.
- 3. x = 30°, 150° or 270°. (2 sin x − 1)(sin x + 1) = 0, so sin x = 1/2 or sin x = −1.
- 4. x = 60° or x = 300°. cos x = 1/2.