Worked example 1
Use the Pythagorean identity
Given sin x = and x is acute, find cos x exactly
- Substitute into sin² x + cos² x = 1
- Solve cos² x = 1 − = .
- Choose the positive root because x is acute.
Answer: cos x =
Higher Mathematics
Higher Maths Exact values and identities
Practise exact values and identities for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Use exact trigonometric values and the identity sin² x + cos² x = 1.
Before you start
- Know the exact values for 0°, 30°, 45°, 60° and 90°.
- Recognise the Pythagorean identity.
- Use quadrant information to choose the correct sign.
Explanation
Exact trigonometric values keep answers in surd or fractional form, avoiding premature rounding.
The identity sin² x + cos² x = 1 lets you recover one ratio from another, but the quadrant determines its sign.
Method and rules
- sin² x + cos² x = 1
- sin 45° = cos 45° =
- tan x = , when cos x is not zero
Worked examples
Worked example 2
Method check
State cos 60° exactly.
- Use the standard exact-value triangle.
So:
Watch out
- Taking both signs without using the stated quadrant.
- Rounding an exact surd or fraction.
- Confusing tan x with
Exam reminder
For exact values and identities, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
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