Solve simple trigonometric equations within a stated range using the basic angle and the quadrant rule

What this dot point is asking

SEAB wants you to solve a trigonometric equation such as $\sin x = 0.5$ and find every solution within a stated range (often $0^\circ$ to $360^\circ$). The method is always the same: find the basic angle from the positive value, decide which quadrants the solutions lie in from the sign, then write down each angle in range. The most common error is giving only the calculator answer and missing the others.

The answer

Step one: find the basic angle

The basic angle (reference angle) is the acute angle you get by taking the inverse trig function of the positive value:

Always use the positive value here; the sign of the original number is handled separately by the quadrants. For example, for $\cos x = -0.5$ the basic angle is $\cos^{-1}(0.5) = 60^\circ$.

Step two: choose the quadrants from the sign

The sign of the right-hand side tells you which quadrants the solutions are in, using "All, Sine, Tangent, Cosine":

• $\sin x$ positive: quadrants 1 and 2; negative: quadrants 3 and 4.
• $\cos x$ positive: quadrants 1 and 4; negative: quadrants 2 and 3.
• $\tan x$ positive: quadrants 1 and 3; negative: quadrants 2 and 4.

Step three: build the angles in each quadrant

With basic angle $\alpha$, the angle in each quadrant is:

• Quadrant 1: $\alpha$
• Quadrant 2: $180^\circ - \alpha$
• Quadrant 3: $180^\circ + \alpha$
• Quadrant 4: $360^\circ - \alpha$

Write down the two (or more) that fall in the chosen quadrants and lie within the given range.

Handling a wider range or a multiple angle

If the range is larger (say up to $720^\circ$), keep adding $360^\circ$ to each solution while it stays in range. If the equation is in a multiple angle like $\sin 2x$, solve for $2x$ over the stretched range first, then divide each answer by $2$.

Examples in context

Example 1. After applying an identity. A question may first need an identity to reduce $2\sin^2 x = 1$ to $\sin x = \pm\tfrac{1}{\sqrt{2}}$, which then splits into two standard equations. So the identity work and the equation-solving method combine in fuller questions.

Example 2. Modelling periodic motion. The height of a point on a turning wheel, or the depth of a tide, follows a sine model. Finding the times when the height takes a particular value means solving a trigonometric equation over a time range, which is exactly this skill applied to a real periodic situation.

Try this

Q1. Solve $\cos x = 0.5$ for $0^\circ \le x \le 360^\circ$. [2 marks]

• Cue. Basic angle $60^\circ$; cosine positive in quadrants 1 and 4, so $x = 60^\circ$ and $x = 300^\circ$.

Q2. Solve $\tan x = 1$ for $0^\circ \le x \le 360^\circ$. [2 marks]

• Cue. Basic angle $45^\circ$; tangent positive in quadrants 1 and 3, so $x = 45^\circ$ and $x = 225^\circ$.

Q3. Solve $\sin x = 0$ for $0^\circ \le x \le 360^\circ$. [2 marks]

• Cue. $\sin x = 0$ at $x = 0^\circ$, $180^\circ$ and $360^\circ$.