Double-angle identities and derived forms | Year 11

01 / LEARN

Understand the idea.

Set the angles equal

The double-angle identities follow by setting both angles equal in the compound-angle formulas. For cosine, use ${sin}^{2} x + {cos}^{2} x = 1$ to obtain alternative forms. Choose the form containing the ratio already known to avoid unnecessary square roots.

The tangent double-angle formula requires $cos x \neq 0$ and $1 - {tan}^{2} x \neq 0$ . These ensure that its original tangent and final quotient exist. If either condition fails, return to sine and cosine to examine the doubled-angle tangent directly.

cos 2 x = 1 - 2 {sin}^{2} x = 2 {cos}^{2} x - 1

Rearrange to handle squares

Rearranging the cosine double-angle forms expresses squared ratios using $cos 2 x$ . This reduces a squared trigonometric expression to a first-power one and also produces half-angle values by choosing the doubled angle appropriately.

A square identity supplies a magnitude, not a sign. Before taking a square root for a half-angle, locate that half-angle in its quadrant. Preserve the exact surd unless the question requests a decimal approximation.

\begin{aligned} {sin}^{2} x & = \frac{1 - cos 2 x}{2} \\ {cos}^{2} x & = \frac{1 + cos 2 x}{2} \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Example 1

For acute $x$ with $sin x = 5 / 13$ , find $sin 2 x$ and $cos 2 x$ .

Worked solution

1
Find the missing cosine.
$cos x = \sqrt{1 - {(\frac{5}{13})}^{2}} = \frac{12}{13}$
The angle is acute, so cosine is the positive square root.
2
Calculate the doubled-angle sine.
$sin 2 x = 2 sin x cos x = 2 \frac{5}{13} \frac{12}{13} = \frac{120}{169}$
The sine double-angle identity uses both original ratios.
3
Choose the cosine form using the given sine.
$cos 2 x = 1 - 2 {sin}^{2} x = 1 - \frac{50}{169} = \frac{119}{169}$
This cosine form avoids introducing any new unknown ratio.

Answer

$sin 2 x = 120 / 169$ and $cos 2 x = 119 / 169$ .

EXAMPLE 02

Example 2

Find $sin (\frac{π}{8})$ exactly.

Worked solution

1
Apply the square identity to the half-angle.
${sin}^{2} (\frac{π}{8}) = \frac{1 - cos (\frac{π}{4})}{2}$
Twice $\frac{π}{8}$ is $\frac{π}{4}$ , whose cosine is known exactly.
2
Simplify the squared value.
${sin}^{2} (\frac{π}{8}) = \frac{1 - \sqrt{2} / 2}{2} = \frac{2 - \sqrt{2}}{4}$
A common denominator puts the expression into a single exact fraction.
3
Choose the sign using the quadrant.
$sin (\frac{π}{8}) = \frac{\sqrt{2 - \sqrt{2}}}{2}$
The half-angle is acute, so its sine is positive.

Answer

$\sqrt{2 - \sqrt{2}} / 2$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $1 - 2 {sin}^{2} x$ .

Show answer & reasoning

$cos 2 x$ .

The cosine double-angle identity is $cos 2 x = 1 - 2 {sin}^{2} x$ , so the expression simplifies directly to $cos 2 x$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

If $cos x = \frac{3}{5}$ , calculate $cos 2 x$ .
2 marks
If $tan x = \frac{1}{2}$ , calculate $tan 2 x$ .
2 marks
Rewrite ${cos}^{2} (3 x)$ using a multiple-angle cosine.
2 marks
Find $cos (\frac{π}{8})$ exactly.
2 marks
Solve $cos 2 x = 1$ on $[0, 2 π)$ .
2 marks
For $x$ in quadrant II with $sin x = \frac{3}{5}$ , find $sin 2 x$ .
2 marks

Read the worked solutions

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

If $cos x = \frac{3}{5}$ , calculate $cos 2 x$ .
2 marks
$2 (\frac{9}{25}) - 1 = - \frac{7}{25}$ .
$- \frac{7}{25}$ .
If $tan x = \frac{1}{2}$ , calculate $tan 2 x$ .
2 marks
$\frac{2 (\frac{1}{2})}{1 - \frac{1}{4}} = \frac{4}{3}$ .
$\frac{4}{3}$ .
Rewrite ${cos}^{2} (3 x)$ using a multiple-angle cosine.
2 marks
Apply the square identity with angle $3 x$ .
$\frac{1 + cos 6 x}{2}$ .
Find $cos (\frac{π}{8})$ exactly.
2 marks
Use $\frac{1 + cos (\frac{π}{4})}{2}$ for its square, then the positive root.
$\frac{\sqrt{2 + \sqrt{2}}}{2}$ .
Solve $cos 2 x = 1$ on $[0, 2 π)$ .
2 marks
$2 x = 2 k π$ gives $x = k π$ .
$x = 0, π$ .
For $x$ in quadrant II with $sin x = \frac{3}{5}$ , find $sin 2 x$ .
2 marks
Cosine is $- \frac{4}{5}$ ; multiply $2 (\frac{3}{5}) (- \frac{4}{5})$ .
$- \frac{24}{25}$ .

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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.

Set the angles equal

Rearrange to handle squares

See the method in action.

Example 1

Find the missing cosine.

Calculate the doubled-angle sine.

Choose the cosine form using the given sine.

Example 2

Apply the square identity to the half-angle.

Simplify the squared value.

Choose the sign using the quadrant.

Try it yourself.

Check your understanding

Keep practising