# Applications of trigonometry for OCR A-level Maths

1. Area of an irregular triangle
2. Sine rule
3. Cosine rule
4. Trigonometric proof

The area of any triangle can be found when an angle and the lengths of the two sides subtending it are known. Given that C is the angle and the sides are a and b, the area of the triangle is given by 1/2 × absinC.

The sine rule states that the ratio of the sine value of an angle and the length of the side opposite it is equal for all sides of a triangle. This is given as sinA/a = sinB/b = sinC/c, which is equivalent to a/sinA = b/sinB = c/sinC. The first version is usually used to find angles, whereas the second version is usually used to find lengths of sides.

In a triangle with a side a opposite an angle A and two other sides b and c, the cosine rules states that: a² = b² + c² − 2bccosA. This can be rearranged to cosA = (b² + c² − a²) ÷ 2bc to find angles more easily.

Trigonometric identities can be used to prove further identities. Trigonometric proofs are performed by starting on one side of the identity and applying known trigonometric identities until the result on the other side of the identity is arrived to.

# 1

Prove that 1 − cos²x − 2sin⁴x = cos2xsinx²x.

LHS = 1 − cos²x − 2sin⁴x
= sin²x − 2sin⁴x
= sin²x(1 − 2sin²x)
= sin²xcos2x
= cos2xsin²x = RHS

Use double angle formula for cosine and the identity sin²x + cos²x = 1.

# 2

A triangle has an angle of size 40° and an angle of 70°. Given that the sides subtending the unknown angle have lengths 8 cm and 10 cm, calculate the area of the triangle.

The sum of the angles in a triangle is 180°.
40° + 70° + θ = 180°
θ = 180° − 70° − 40°
θ = 70°

Area = 1/2 × absinC
Area = 1/2 × 8 cm × 10 cm × sin(70°)
Area = 37.6 cm² (to 3 s. f.)

37.6 cm²

# 3

Zain is making a garland out of triangles. Given that each triangle has two sides of 8 cm and 10 cm, and an angle of 60° opposite a side of unknown length, calculate the length of the third side.

By the cosine rule, a² = b² + c² − 2bccosA.
a² = 10² + 8² − 2 × 10 × 8 × cos(60°)
a² = 84
a = √74.4 = 2√21 cm

2√21 cm

# 4

Find the area of the given triangle.

By the cosine rule, cosA = (b² + c² − a²) ÷ 2bc.
cosA = (8² + 9² − 12²) ÷ (2 × 8 × 9)
cosA = 1/144
a = arccos(1/144) = 89.6° (to 3 s. f.)

Area = 1/2 × absinC
Area = 1/2 × 8 cm × 9 cm × sin(89.6°)
Area = 36.0 cm² (to 3 s. f.)

36.0 cm²

# 5

An angle of size 45° is subtended by a side of length 6 cm and a side of unknown length in a triangle. Given that the area of the triangle is 3√2 cm², find the unknown length.

Let x be the unknown length.
Area = 1/2 × absinC
3√2 cm² = 1/2 × 6 cm × x × sin(45)
x = 2 cm

2 cm

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