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.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

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

Skill quiz: Choose a right-triangle ratio

Identify the sides involved before applying sine, cosine or tangent.

Trigonometric functions and trianglesPremium

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

Skill quiz: Match sides with opposite angles

Use the sine and cosine laws quiz and focus on matching each side to its opposite angle.

The sine and cosine lawsPremium

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

Skill quiz: Use two sides and the included angle

Choose the cosine law when appropriate and keep the subtraction term in the calculation.

The sine and cosine lawsPremium

Where these appear

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

Skill quizzes: Use trigonometry in larger diagrams

Identify useful triangles in circle problems, then progress to more complex geometric shapes.

Common mistakes

Using $SOH - CAH - TOA$ on a triangle that is not right-angled.
Matching the wrong side with the wrong opposite angle in the sine law.
Forgetting the minus sign in the cosine law.

Skill quizzes: Choose the law for the given information

Check whether the triangle is right-angled and match each side and angle before applying a formula.