Compound-angle identities | Year 11

01 / LEARN

The idea, step by step

Combine angles using products

Trigonometric functions do not distribute over addition. Sine of a sum involves a sine–cosine product and a cosine–sine product. Cosine of a sum subtracts a sine–sine product from a cosine–cosine product. Match the formula and signs before substituting.

Choose component angles with known exact values: $75^{\circ} = 45^{\circ} + 30^{\circ}$ or $15^{\circ} = 45^{\circ} - 30^{\circ}$ . Keep surds exact throughout the calculation.

sin (A \pm B) = sin A cos B \pm cos A sin B

Recover missing information first

If one ratio is given, find the other with ${sin}^{2} A + {cos}^{2} A = 1$ and choose its sign using the stated quadrant. Without a quadrant, the answer may not be unique. Draw a small reference triangle or locate the point on the unit circle.

The tangent compound-angle rule follows by dividing the sine formula by the cosine formula. Its denominator must be nonzero and the original tangent values must exist. An undefined tangent must not be reported as a very large finite answer.

cos (A \pm B) = cos A cos B \mp sin A sin B

A common trap

Writing $sin (A + B) = sin A + sin B$ .

Instead: Use the compound-angle identity containing two products.

02 / WORKED EXAMPLES

See the thinking

Example 1

Calculate $sin 75^{\circ}$ exactly.

Write the angle as $45^{\circ} + 30^{\circ}$ .
Substitute $(\sqrt{2} / 2) (\sqrt{3} / 2) + (\sqrt{2} / 2) (1 / 2)$ .
Combine the terms over 4.

(\sqrt{6} + \sqrt{2}) / 4

Example 2

For acute $A, B$ , $sin A = 3 / 5$ and $cos B = 12 / 13$ . Find $cos (A + B)$ .

Acute angles give $cos A = 4 / 5$ , $sin B = 5 / 13$ .
Use $cos A cos B - sin A sin B$ .
Calculate $48 / 65 - 15 / 65 = 33 / 65$ .

33 / 65

Check your understanding

Calculate $cos (45^{\circ} - 30^{\circ})$ exactly.

Show answer

$(\sqrt{6} + \sqrt{2}) / 4$ .

For cosine of a difference, add the cosine–cosine and sine–sine products.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $sin 15^{\circ}$ exactly.
2 marks
Find $cos 75^{\circ}$ exactly.
2 marks
Find $tan 75^{\circ}$ exactly.
2 marks
For acute $A, B$ with $sin A = 5 / 13$ and $sin B = 3 / 5$ , find $sin (A + B)$ .
2 marks
Given $tan A = 2$ , $tan B = 1 / 3$ , calculate $tan (A - B)$ .
2 marks
Disprove $cos (A + B) = cos A + cos B$ .
2 marks

Read the worked solutions

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

Find $sin 15^{\circ}$ exactly.
2 marks
Expand $sin (45^{\circ} - 30^{\circ})$ .
$(\sqrt{6} - \sqrt{2}) / 4$ .
Find $cos 75^{\circ}$ exactly.
2 marks
Expand $cos (45^{\circ} + 30^{\circ})$ .
$(\sqrt{6} - \sqrt{2}) / 4$ .
Find $tan 75^{\circ}$ exactly.
2 marks
Use $(1 + 1 / \sqrt{3}) / (1 - 1 / \sqrt{3})$ , then rationalise.
$2 + \sqrt{3}$ .
For acute $A, B$ with $sin A = 5 / 13$ and $sin B = 3 / 5$ , find $sin (A + B)$ .
2 marks
The missing cosines are $12 / 13$ and $4 / 5$ .
Add $20 / 65 + 36 / 65$ .
$56 / 65$ .
Given $tan A = 2$ , $tan B = 1 / 3$ , calculate $tan (A - B)$ .
2 marks
$(2 - 1 / 3) / (1 + 2 / 3) = (5 / 3) / (5 / 3) = 1$ .
1.
Disprove $cos (A + B) = cos A + cos B$ .
2 marks
Both sides are defined, and the unequal values disprove the universal claim.
Set $A = B = 0$ : the two sides are 1 and 2.

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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 ↗

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