Solving Trigonometric Equations

Trig equations are where 0606 students lose marks without being wrong, every value they find is correct; they just don’t find them all. The method below is a completeness machine.

The five-step routine

Solve $2\sin^2 x - \sin x - 1 = 0$ for $0^\circ \le x \le 360^\circ$.

1. Range and units first. Degrees, full circle, expect multiple solutions.
2. Reduce to single functions. Factorise the quadratic in $\sin x$: $(2\sin x + 1)(\sin x - 1) = 0$ $\to$ $\sin x = -\frac{1}{2}$ or $\sin x = 1$
3. Reference angle for each. $\sin x = -\frac{1}{2}$: reference $30^\circ$; sine negative $\to$ third and fourth quadrants (unit circle)
4. Sweep the full range. $x = 180^\circ + 30^\circ = 210^\circ$, $x = 360^\circ - 30^\circ = 330^\circ$. And $\sin x = 1$ $\to$ $x = 90^\circ$.
5. Collect: $x = 90^\circ, 210^\circ, 330^\circ$, count against the graph’s expectations.

Mixed-function equations get an identity substitution first ($\cos^2 x \to 1 - \sin^2 x$; or $\div\cos x$ turning $\sin x = \cos x$ into $\tan x = 1$).

The two cardinal sins

Dividing by a trig factor. $\sin x \cos x = \cos x \div \cos x$ deletes every solution of $\cos x = 0$. Move and factorise instead: $\cos x(\sin x - 1) = 0$, both families survive. Any division by something that can be zero is solution destruction.

Stopping at the calculator’s answer. $\sin^{-1}(-0.5)$ returns $-30^\circ$, one value, often outside your range. The calculator gives the reference information; the unit circle gives the solutions.

Compound angles: expand the range

For $\sin(2x - 30^\circ) = \frac{1}{2}$ with $0^\circ \le x \le 180^\circ$: the argument $u = 2x - 30^\circ$ runs over $-30^\circ \le u \le 330^\circ$, solve for $u$ across that whole window ($u = 30^\circ, 150^\circ$), then convert back ($x = 30^\circ, 90^\circ$). Solving for $x$ directly, or forgetting the doubled window, halves your solution set, the most common compound-angle error. A multiplier $b$ inside the function means roughly $b$ times as many solutions: expect them.

Radian ranges

$0 \le x \le 2\pi$ demands radian answers, exact ($\frac{\pi}{6}, \frac{5\pi}{6}$) when the values are standard, calculator in radian mode when not. Converting the range to degrees “to be safe” invites conversion slips and costs time; work native.

Common mistakes

• Solutions outside or missing from the stated range
• Division deleting a solution family
• Second-quadrant (or third/fourth) partners omitted
• Compound arguments solved over the unexpanded range
• Degree answers to radian ranges
• The quadratic structure unfactorised (treating $2\sin^2 x - \sin x - 1$ as unsolvable)

Full topic context: Trigonometry notes · the full exam drill: trig technique guide.