Solve trigonometric equations within a stated interval, finding the basic angle and using symmetry to obtain every solution

What this dot point is asking

SEAB wants you to solve equations such as $\sin\theta = \tfrac{1}{2}$ or $2\cos^2\theta + \cos\theta - 1 = 0$ and to list every solution inside a stated interval (usually $0^\circ$ to $360^\circ$, or in radians). The key is to find the basic angle once, then use the symmetry of the trigonometric functions to find all the angles that share that ratio.

The answer

The basic angle

The basic angle is the acute angle whose sine, cosine or tangent has the same magnitude as the value you want. Find it from the inverse function applied to the positive value: for $\sin\theta = \tfrac{1}{2}$, the basic angle is $\sin^{-1}\tfrac{1}{2} = 30^\circ$.

Using quadrant symmetry

From the basic angle $\beta$, the solutions in $0^\circ$ to $360^\circ$ are:

• $\sin\theta = +$: $\beta$ and $180^\circ - \beta$ (first and second quadrants).
• $\sin\theta = -$: $180^\circ + \beta$ and $360^\circ - \beta$.
• $\cos\theta = +$: $\beta$ and $360^\circ - \beta$ (first and fourth).
• $\cos\theta = -$: $180^\circ - \beta$ and $180^\circ + \beta$.
• $\tan\theta = +$: $\beta$ and $180^\circ + \beta$.
• $\tan\theta = -$: $180^\circ - \beta$ and $360^\circ - \beta$.

The sign of the right-hand side tells you which quadrants to use.

Equations in a multiple angle

If the equation is in $2\theta$ or $3\theta$, first widen the range to match (for $2\theta$ over $0^\circ$ to $360^\circ$, work in $0^\circ$ to $720^\circ$), solve for the multiple angle, then divide each solution back down. This recovers solutions you would otherwise miss.

Equations reducible to a quadratic

When an equation mixes $\sin^2\theta$ and $\sin\theta$ (or uses an identity to get there), substitute and treat it as a quadratic in the ratio, solve, then solve each resulting simple equation, rejecting any value outside $[-1, 1]$.

Examples in context

Example 1. Times of a given tide height. A tide model $h = 2 + 3\sin(30t)^\circ$ reaching a set height becomes a sine equation in $30t$; widening the range and solving finds every time in the day the tide is at that level.

Example 2. Phase of an oscillation. After applying the R-formula, an equation $R\sin(\theta + \alpha) = k$ is a single-ratio equation; solving it with the basic angle gives the phase angles at which a combined oscillation reaches a target value.

Try this

Q1. Solve $\cos\theta = \dfrac{1}{2}$ for $0^\circ \leq \theta \leq 360^\circ$. [2 marks]

• Cue. Basic angle $60^\circ$; cosine positive in quadrants one and four: $\theta = 60^\circ$ or $300^\circ$.

Q2. Solve $\tan\theta = -1$ for $0^\circ \leq \theta \leq 360^\circ$. [2 marks]

• Cue. Basic angle $45^\circ$; tangent negative in quadrants two and four: $\theta = 135^\circ$ or $315^\circ$.

Q3. Solve $\sin\theta = -\dfrac{1}{2}$ for $0^\circ \leq \theta \leq 360^\circ$. [3 marks]

• Cue. Basic angle $30^\circ$; sine negative in quadrants three and four: $\theta = 210^\circ$ or $330^\circ$.