Exact values and trigonometric graphs Cambridge IGCSE Mathematics revision
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In plain words
Sine and cosine aren't only for triangles. Plot them for angles from 0° to 360° and you get waves that repeat for ever. Because they repeat, an equation such as sin x = 0.5 has more than one answer.
Three things to know
- Exact values: sin 30° = cos 60° = ½. sin 60° = cos 30° = √3/2. sin 45° = cos 45° = √2/2. tan 45° = 1, tan 30° = 1/√3 and tan 60° = √3.
- The graphs of sin x and cos x are waves between −1 and 1 that repeat every 360°. The graph of tan x repeats every 180° and has gaps where it shoots off to infinity, at 90° and 270°.
- Between 0° and 360° a trigonometric equation usually has two solutions. If the calculator gives x: for sin the other is 180° − x; for cos it is 360° − x; for tan it is x + 180°.
Worked example
Solve sin x = 0.5 for 0° ≤ x ≤ 360°.
- The calculator gives x = sin⁻¹(0.5) = 30°.
- For sine, the second solution is 180° − 30° = 150°.
- x = 30° or x = 150°.
Tips and tricks
- Sketch the graph and draw a horizontal line at the value: you can see how many solutions there are, and roughly where.
- Sine and cosine are never bigger than 1 or smaller than −1. sin x = 2 has no solutions.
It lands in your notebook with its questions as flashcards.
Exact values and trigonometric graphs: 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 is the exact value of sin 30°?
It equals cos 60°.
What is the greatest value of sin x?
The sine graph stays between −1 and 1.
After how many degrees does the graph of y = sin x repeat?
This is its period.
sin x = sin 40°. What is the other value of x between 0° and 360°?
180° − 40°.
What is the exact value of tan 45°?
The opposite and adjacent sides are equal.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 8 marks. Write your answers on paper, then check them.
Exact values and trigonometric graphs
Cambridge IGCSE Mathematics 0580 · 8 marks · papermunch.org
Name ______________________________ Date ______________
Write down the exact values of cos 60° and tan 45°.[2]
Show answerHide answer
½ and 1.
Solve cos x = 0.5 for 0° ≤ x ≤ 360°.[3]
Show answerHide answer
x = 60° or x = 300°.
Solve tan x = 1 for 0° ≤ x ≤ 360°.[3]
Show answerHide answer
x = 45° or x = 225°.
Answers: Exact values and trigonometric graphs
- 1. ½ and 1.
- 2. x = 60° or x = 300°.
- 3. x = 45° or x = 225°.