What you need to know
Further Pure trigonometry rewards exact values, identity manipulation and complete solution sets. Always pay attention to the requested interval and whether an equation can have more than one solution.
- Know exact values for standard angles.
- Use fundamental identities to rewrite expressions.
- Solve the basic trigonometric equation before generating all solutions in the interval.
- Check units: degrees and radians should never be mixed.
Key mathematics
Pythagorean identity
\sin^2\theta+\cos^2\theta=1
A key identity for converting between sine and cosine.Tangent definition
\tan\theta=\frac{\sin\theta}{\cos\theta}
Useful when rearranging identities and equations.Worked example
Solve $2\sin\theta=1$ for $0^\circ\le\theta\le360^\circ$. First $\sin\theta=\frac12$. Sine is positive in quadrants I and II, so $$\theta=30^\circ,150^\circ.$$
EXAM TECHNIQUE
Exam tips
- Write the requested interval at the top of your working.
- Use the sign of the trig function to identify relevant quadrants.
- For identities, transform one side toward the other instead of changing both sides randomly.
WATCH OUT
Common mistakes
- Giving only the calculator principal value and missing another solution.
- Mixing degree-mode answers with radian intervals.
- Dividing by a trig expression that could be zero and accidentally losing solutions.
Frequently asked questions
Why are there often multiple trig solutions?
Trigonometric functions are periodic, so the same value can occur at more than one angle in the interval.
Should I use exact values?
Use exact values when the angle is standard and the question expects an exact answer.
How should I prove an identity?
Usually work from one side and use known identities until it becomes the other side.