The six trigonometric functions Cambridge IGCSE Additional Mathematics revision
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In plain words
You already know sine, cosine and tangent. Additional Mathematics adds their three reciprocals, each with its own name, and expects all six to work for any angle, not just the ones inside a right-angled triangle.
6 things to know
- sec x = 1 ÷ cos x. cosec x = 1 ÷ sin x. cot x = 1 ÷ tan x, which is also cos x ÷ sin x.
- Exact values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3. sin 45° = cos 45° = 1/√2, tan 45° = 1. sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
- For angles beyond 90°, the size of the answer comes from the related acute angle, and the sign comes from the quadrant.
- Signs: all three are positive from 0° to 90°. Only sine is positive from 90° to 180°. Only tangent from 180° to 270°. Only cosine from 270° to 360°.
- A reciprocal function has the same sign as the function it comes from.
- Given one ratio for an acute angle, draw a right-angled triangle, use Pythagoras to find the third side, and read off the others.
Worked example
Find the exact values of sec 60°, cos 150° and sin 210°.
- sec 60° = 1 ÷ cos 60° = 1 ÷ (1/2) = 2.
- 150° is in the second quadrant, where cosine is negative. Its related acute angle is 30°. cos 150° = −√3/2.
- 210° is in the third quadrant, where sine is negative. Its related acute angle is 30°. sin 210° = −1/2.
Worked example
θ is acute and sin θ = 3/5. Find cos θ, tan θ and sec θ.
- Draw a right-angled triangle with opposite 3 and hypotenuse 5. The adjacent side is √(25 − 9) = 4.
- cos θ = 4/5 and tan θ = 3/4.
- sec θ = 1 ÷ cos θ = 5/4.
Tips and tricks
- The third letter is the clue: sec goes with cos, cosec goes with sin, cot goes with tan.
- Calculators have no sec, cosec or cot keys. Work out cos, sin or tan and then take the reciprocal.
It lands in your notebook with its questions as flashcards.
The six trigonometric functions: 6 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is sec x?
Sec goes with cos.
What is cosec 90°?
1 ÷ sin 90° = 1 ÷ 1.
What is cot x in terms of sin x and cos x?
It is tan x turned upside down.
What is sin 150°?
Related angle 30°, and sine is positive in the second quadrant.
In which quadrant is tangent positive and sine negative?
From 180° to 270°, only tangent is positive.
What is cos 120°?
Related angle 60°, and cosine is negative in the second quadrant.
Quiz
6 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.
The six trigonometric functions
Cambridge IGCSE Additional Mathematics 0606 · 7 marks · papermunch.org
Name ______________________________ Date ______________
State the exact values of cosec 30° and cot 45°.[2]
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cosec 30° = 2 and cot 45° = 1.
θ is acute and cos θ = 5/13. Find the exact values of sin θ, tan θ and sec θ.[3]
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sin θ = 12/13, tan θ = 12/5 and sec θ = 13/5. The third side is 12.
State the exact value of tan 135°.[2]
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−1. The related acute angle is 45°, and tangent is negative in the second quadrant.
Answers: The six trigonometric functions
- 1. cosec 30° = 2 and cot 45° = 1.
- 2. sin θ = 12/13, tan θ = 12/5 and sec θ = 13/5. The third side is 12.
- 3. −1. The related acute angle is 45°, and tangent is negative in the second quadrant.