Solving trigonometric equations
To solve $\sin\theta = k$, $\cos\theta = k$ or $\tan\theta = k$ for $0° \le \theta \le 360°$, first find the reference angle $\alpha$ (the acute angle whose sine, cosine or tangent equals $|k|$). Then use the sign of $k$ to place the solutions in the correct quadrants.
Key formula
Reference angle $\alpha$ satisfies $\sin\alpha = |k|$ (or $\cos\alpha$, $\tan\alpha$). Signs (CAST): $\sin\theta$ is positive in Q1, Q2; $\cos\theta$ is positive in Q1, Q4; $\tan\theta$ is positive in Q1, Q3. Full-turn positions: Q1 $=\alpha$, Q2 $=180°-\alpha$, Q3 $=180°+\alpha$, Q4 $=360°-\alpha$.
Worked example
Worked example
Solve $\cos\theta = -0.5$ for $0° \le \theta \le 360°$. Reference angle: $\cos\alpha = 0.5 \Rightarrow \alpha = 60°$. Cosine is negative in Q2 and Q3, so $\theta = 180° - 60° = 120°$ and $\theta = 180° + 60° = 240°$.
Remember
- The reference angle uses $|k|$; the sign only chooses the quadrants.
- Sine positive $\to$ Q1, Q2; cosine positive $\to$ Q1, Q4; tangent positive $\to$ Q1, Q3.
- Check every answer lies in $0° \le \theta \le 360°$.