Trig identities and equations · Year 13
In Year 13 you learn: double angle formulas.
Trig identities are equations that are true for every angle. The two key ones let you rewrite expressions and solve trig equations, which usually have more than one solution in a given range.
Key facts
- sin²θ + cos²θ = 1.
- tan θ = sin θ ÷ cos θ.
- sin is positive in the 1st and 2nd quadrants; cos in the 1st and 4th; tan in the 1st and 3rd.
- Solutions of sin θ = k: θ and 180° − θ; of cos θ = k: θ and 360° − θ; of tan θ = k: θ and θ + 180°.
- For equations like sin 2θ = k, solve for 2θ over the doubled range, then halve.
- Quadratic in sin or cos: factorise, then solve each bracket.
Worked examples
Example 1
Solve 2 sin θ = −1 for 0° ≤ θ ≤ 360°.
Answer: θ = 210° or 330°
Example 2
Solve 2cos²x − cos x − 1 = 0 for 0° ≤ x < 360°.
Answer: x = 0°, 120° or 240°