Sine Rule and Cosine Rule

 

In this O-level E-Math revision note, you will learn the Sine Rule and Cosine Rule and how to apply them to solve common Trigonometry exam questions.

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math revision notes, sine rule, cosine rule

 

Sine Rule

For any triangle ABC,

a/sin A= b/sin B = c/ sin C

or

sin A /a = sin B/ b = sin C/ c

where A, B and C are the interior angles, and a, b and c are lengths of their opposite sides respectively.

sine rule

Note: Never use Sine Rule on right-angled triangles as we are able to apply the basic trigonometric ratios!

When do we apply Sine Rule?

(1) identify the unknown angle (or side) and its known opposite side (or angle).
(2) identify another pair of known angle and its known opposite side.
(3) apply Sine Rule and write the unknown in the numerator of the formula.

 

Sine Rule Example 1 (given 2 angles and 1 side)

Given angle B = 50°, angle C = 26° and AC = 12 cm, find the length of AB.

sine rule example 1

Applying sine rule,

AB/ sin B = AC/ sin C

AB/ sin 50° = 12/ sin 26°

AB = 12 sin 50°/sin 26°

= 21.0cm (3 sf)

 

Sine Rule Example 2 (given 2 sides and an angle)

Given angle C = 36°, AB = 9 cm and BC = 15 cm, find acute angle A.

sine rule example 2

Applying sine rule,

sin A/BC = sin C/ AB

sin A/15 = sin 36°/9

sin A = 15 sin 36°/9

= 0.9796 (4 dp)

A = sin-1 0.9796

= 78.4° (1 dp)

Note: When applying the Sine Rule to find an unknown angle, always check if the angle you are finding is acute or obtuse!

In the diagram above, it is clear that the angle is acute, so we can ignore the obtuse angle answer.

Cosine rule

cosine rule

For any triangle ABC,

a2 = b2 + c2 – 2bc cos A

b2 = a2 + c2 – 2ac cos B

c2 = a2 + b2 – 2ab cos C

or 

cos A = (b2 + c2 a2)/2bc

cos B = (a2 + c2 b2)/2ac

cos C = (a2 + b2 c2)/2ab

where A, B and C are the interior angles within the triangle, and a, b and c are lengths of their opposite sides respectively.

Note that you only need to memorise

a2 = b2 + c2 – 2bc cos A

and

cos A = (b2 + c2 a2)/2bc

as the rest of the formulas can be derived easily.

When do we apply Cosine Rule?

To find an unknown side of a triangle:

To find an unknown angle in a triangle:

Condition: When an angle and the two sides that form the angle are known, apply Cosine Rule to find the unknown side that is opposite the angle.

Step 1: Use a2 = b2 + c2 – 2bc cos A

Step 2: Substitute values of b, c and A.

Step 3: Apply square root to both sides of
the equation.

Note: Ignore the negative version since it is
understood that length measurements cannot be negative.

Condition: When all three sides of a triangle are known, apply Cosine Rule to find any unknown angle within the triangle.

Step 1: Use cos A = (b2 + c2 a2)/2bc

Step 2: Substitute values of a, b and c.

Step 3: Solve for angle A by applying
inverse cosine to both sides of the equation.

Note: There is only 1 angle, unlike the
ambiguous cases when we apply sine
inverse

Cosine Rule Example 1 (given 2 sides and the known angle formed by the 2 sides)

Given AB = 8 cm, BC = 5 cm and angle B = 48°, find the length of AC.

cosine rule example 1

b2 = a2 + c2 – 2ac cos B

AC2 = 52 + 82 – 2(5)(8) cos 48°

AC =√ (52 + 82 – 2(5)(8) cos 48°)

= 5.9556(4 dp)

= 5.96 cm (3 sf)

 

Cosine Rule Example 2 (given 3 sides)

The diagram below shows ∆ABC. Given AB = 24 cm, BC = 34 cm and AC = 30 cm, find the value of θ.

cosine rule example 2

cos θ = (b2 + c2 a2)/2bc

= (302 + 242 – 342)/2(30)(24)

θ = cos-1( (302 + 242 – 342)/2(30)(24) )

= 77.1604°

= 77.2° (1 dp)

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