Solving trig equations and describing all solutions | Year 11

01 / LEARN

Understand the idea.

Principal value is only a start

An inverse trigonometric function returns one principal value. To solve an equation, also use unit-circle symmetry and periodicity to find every required branch. A positive sine value normally occurs in quadrants I and II within one cycle.

Transform the interval along with the argument. If $0 \leq x < 2 π$ and $u = 2 x$ , then $0 \leq u < 4 π$ . Solve across those two argument cycles before dividing by two, and check whether each interval endpoint is included.

\begin{array}{c} sin x = sin α \\ ⟹ x = α + 2 k π \\ or x = π - α + 2 k π \\ k \in ℤ \end{array}

Factor and preserve restrictions

Factor before dividing by an unknown sine or cosine. In a zero product, either factor may vanish; division could remove a legitimate branch. An equation containing denominators also requires their original zeros to remain excluded.

For a quadratic in sine or cosine, substitute a single variable and solve the algebraic equation. Reject roots outside $[- 1, 1]$ , then solve each surviving trigonometric equation on the full required interval. Finally check the candidates in the original equation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Example 1

Solve $2 sin x = 1$ on $[0, 2 π)$ .

Worked solution

1
Isolate sine.
$2 sin x = 1 \Rightarrow sin x = \frac{1}{2}$
Dividing by the nonzero constant two preserves every solution.
2
Find the two unit-circle branches.
$x = \frac{π}{6} or x = π - \frac{π}{6}$
Sine is positive in quadrants I and II, with reference angle $\frac{π}{6}$ .
3
Apply the stated interval.
$x = \frac{π}{6}, \frac{5 π}{6}$
Both values lie in $[0, 2 π)$ ; neither endpoint is an additional solution.

Answer

$x = \frac{π}{6}, \frac{5 π}{6}$ .

EXAMPLE 02

Example 2

Solve $cos (2 x) = 0$ on $[0, 2 π)$ .

Worked solution

1
Transform the argument and interval.
$u = 2 x, 0 \leq u < 4 π$
The argument completes two cycles while $x$ traverses the stated interval.
2
List every cosine zero in that interval.
$u = \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, \frac{7 π}{2}$
Cosine vanishes at odd multiples of $\frac{π}{2}$ .
3
Convert each value back to x.
$x = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$
Dividing all four argument values by two retains every solution in $[0, 2 π)$ .

Answer

$x = \frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Give every real solution of $tan x = 1$ .

Show answer & reasoning

$x = \frac{π}{4} + k π$ , $k \in ℤ$ .

Tangent equals $1$ at $\frac{π}{4}$ and repeats every $π$ . Thus all real solutions are $x = \frac{π}{4} + k π$ , with $k \in ℤ$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Solve $cos x = - \frac{1}{2}$ on $[0, 2 π)$ .
2 marks
Solve $sin x = 0$ on $[- π, π]$ .
2 marks
Give all real solutions of $cos x = 1$ .
2 marks
Solve $2 {cos}^{2} x - cos x - 1 = 0$ on $[0, 2 π)$ .
2 marks
Solve $sin (2 x) = 0$ on $[0, π]$ .
2 marks
Solve $sin x (2 cos x - 1) = 0$ on $[0, 2 π)$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Solve $cos x = - \frac{1}{2}$ on $[0, 2 π)$ .
2 marks
Reference angle $\frac{π}{3}$ ; use quadrants II and III.
$\frac{2 π}{3}, \frac{4 π}{3}$ .
Solve $sin x = 0$ on $[- π, π]$ .
2 marks
All integer multiples of $π$ in the closed interval count.
$- π, 0, π$ .
Give all real solutions of $cos x = 1$ .
2 marks
Cosine is 1 at complete turns.
$x = 2 k π$ , $k \in ℤ$ .
Solve $2 {cos}^{2} x - cos x - 1 = 0$ on $[0, 2 π)$ .
2 marks
Factor $(2 cos x + 1) (cos x - 1) = 0$ ; solve cosine $- \frac{1}{2}$ or 1.
$0, \frac{2 π}{3}, \frac{4 π}{3}$ .
Solve $sin (2 x) = 0$ on $[0, π]$ .
2 marks
$2 x = k π$ with $k = 0, 1, 2$ .
$0, \frac{π}{2}, π$ .
Solve $sin x (2 cos x - 1) = 0$ on $[0, 2 π)$ .
2 marks
Sine zero gives 0 and $π$ ; cosine $\frac{1}{2}$ gives the other two.
$0, \frac{π}{3}, π, \frac{5 π}{3}$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Further Trigonometry: periodic functions and models, trigonometric identities and reciprocal functions; the Library sequence follows the approved school-topic structure.

Official curriculum source ↗

In this lesson

Understand the idea.

Principal value is only a start

Factor and preserve restrictions

See the method in action.

Example 1

Isolate sine.

Find the two unit-circle branches.

Apply the stated interval.

Example 2

Transform the argument and interval.

List every cosine zero in that interval.

Convert each value back to x.

Try it yourself.

Check your understanding

Keep practising