AHL 3.8 (HL)—Unit circle and trigonometric equations

Solving a trigonometric equation means finding every angle in the stated interval that produces the target value. Periodicity creates repeated solutions.

Use the unit circle or graph to identify the reference angle and the quadrants with the correct sign, then add the function's period. A calculator's principal value is only one solution unless the interval makes it complete.

For sin x=0.5 on 0≤x≤2π, x=π/6 and 5π/6. Stopping at π/6 misses the second intersection of the sine curve with 0.5.

The number of solutions depends on the interval and period. Check endpoints and do not use the cosine rule for a sine equation.

Unit-circle definitions are $\cos heta=x$ and $\sin heta=y$ for the point $(x,y)$ on the unit circle, so $\cos^2 heta+\sin^2 heta=1$ and $\tan heta=\sin heta/\cos heta$ when $\cos heta\ne0$. The sine rule's ambiguous case can give two triangles because $\sin heta=\sin(\pi- heta)$; accept only solutions that satisfy the side lengths and stated interval. Exact trig values are helpful but are not directly assessed.