Year 10: Triangles, sine law and cosine law

Right-angled triangle trigonometry is the starting point, but Year 10 also introduces the sine law and cosine law for non-right-angled triangles. These tools let you solve many geometry problems involving circles and more complex diagrams.

Use

SOH - CAH - TOA

for right-angled triangles, the sine law when you have a matching side-angle pair, and the cosine law for two sides with the included angle or for three sides.

Right-angled triangle review

Example 1: In a right-angled triangle, the angle is

40 °

and the adjacent side is

7 cm

. Find the opposite side.

Use $\tan θ = opposite / adjacent$ .
$\tan 40 ° = x / 7$ .
$x = 7 \tan 40 ° \approx 5.87 cm$ .

The sine law

The sine law is useful when a side is matched with its opposite angle.

Example 2: In triangle

A B C

, side

a = 8 cm

A = 40 °

, and

B = 65 °

. Find side

b

Use $a / \sin A = b / \sin B$ .
$8 / \sin 40 ° = b / \sin 65 °$ .
$b = 8 \sin 65 ° / \sin 40 °$ .
$b \approx 11.28 cm$ .

The cosine law

The cosine law is especially useful when you know two sides and the included angle.

Example 3: Two sides of a triangle are

9 cm

and

12 cm

with included angle

60 °

. Find the third side.

Use $c^{2} = a^{2} + b^{2} - 2 a b \cos C$ .
$c^{2} = 9^{2} + 12^{2} - 2 \times 9 \times 12 \times \cos 60 °$ .
$c^{2} = 81 + 144 - 108 = 117$ .
$c = \sqrt 117 \approx 10.82 cm$ .

Where these appear

These ideas appear in circle problems, navigation, survey-style questions, and complex geometric diagrams where drawing an auxiliary triangle helps.

Common mistakes