Year 12 Trigonometry: Solving trigonometric equations on intervals

Solving a trigonometric equation on a fixed interval is one of the most common Year 12 tasks. The core method is consistent: isolate the trigonometric ratio, identify the reference angle, use quadrant logic to locate every solution on the interval, then check whether any algebraic manipulation introduced a value that should be rejected. The challenge is usually not the first solution. It is finding all valid solutions and stopping at the correct place once the interval has been respected.

Solve the trig relationship first, then use the interval as a filter on the possible angles.

Standard interval method

Rearrange until one trigonometric function is isolated if possible.
Find the reference angle from an exact value or calculator value.
Use the sign of the function to choose the correct quadrants.
List every solution on the stated interval, not just the principal angle.

Worked example 1

Problem: Solve

2 \sin (x) = 1

for

0 \leq x < 2 pi

Rearrange: $\sin (x) = 1 / 2$ .
The reference angle is $pi / 6$ .
Sine is positive in quadrants 1 and 2.
So the solutions are $x = pi / 6$ and $x = 5 pi / 6$ .

Answer:

x = pi / 6

x = 5 pi / 6

Worked example 2

Problem: Solve

2 \cos^{2} (x) - 3 \cos (x) + 1 = 0

for

0 \leq x < 2 pi

Treat the equation as a quadratic in $\cos (x)$ .
Factor: $(2 \cos (x) - 1) (\cos (x) - 1) = 0$ .
So either $\cos (x) = 1 / 2$ or $\cos (x) = 1$ .
For $\cos (x) = 1 / 2$ , the reference angle is $pi / 3$ , and cosine is positive in quadrants 1 and 4. This gives $x = pi / 3$ and $x = 5 pi / 3$ .
For $\cos (x) = 1$ , the only solution on the interval is $x = 0$ .

Answer:

x = 0

pi / 3

, and

5 pi / 3

Why interval questions go wrong

Most interval mistakes come from finishing too early. A calculator often gives one principal angle, but that is only the starting point. The interval question is really asking for every point on the graph of the trig function inside the specified domain where the condition is satisfied. That means your answer should look like a small complete set, not a single isolated angle unless the geometry of the question truly forces that.

Worked example 3

Problem: Solve

\tan (x) = - 1

for

- pi < x \leq pi

The reference angle is $pi / 4$ .
Tangent is negative in quadrants 2 and 4.
On $0 \leq x < 2 pi$ , those would be $3 pi / 4$ and $7 pi / 4$ .
Now rewrite them on the required interval $- pi < x \leq pi$ .
$3 pi / 4$ stays as it is, while $7 pi / 4$ becomes $- pi / 4$ .

Answer:

x = - pi / 4

x = 3 pi / 4

Common traps

Giving the principal value only.
Using the wrong quadrants for the sign of the function.
Ignoring that the interval may include negative angles.
Forgetting to factor carefully when the equation is quadratic in sine or cosine.

Revision focus

The strongest revision approach is to write the interval at the top of the page before you do any algebra. That keeps the real goal visible. It also helps to sketch the unit circle and mark candidate solutions as you go, because the diagram shows immediately whether you have found one, two, or more answers.

Another good habit is to substitute each final answer back into the original equation. This is quick, and it catches sign mistakes and missed restrictions before they cost marks in longer problem-solving questions.

Practice links

General solution of trigonometric equations