 # Trigonometric functions for AQA GCSE Maths 1. Trigonometric functions
2. Exact values of trigonometric functions in degrees

Sine, cosine and tangent are trigonometric ratios used to find angles and sides in a right-angled triangle. The sine of an angle can be calculated by dividing the opposite side to the angle by the hypotenuse. The cosine of an angle can be calculated by dividing the adjacent side to the angle by the hypotenuse. The tangent can be calculated by dividing the opposite side to the angle by the adjacent one. the three trigonometric ratios are usually given as sin, cos and tan. To find angles in a right-angled triangle, the inverse of these trigonometric ratios has to be used. The values for some specific trigonometric ratios are useful to know. The exact values of sine for the angles 0, 30°, 45°, 60° and 90° are are 0, 1/2, 1/√2, √3/2 and 1 respectively. The exact values of cosine for the angles 0, 30°, 45°, 60° and 90° are are 1, √3/2, 1/√2, 1/2 and 1 respectively. The exact values of tangent for the angles 0, 30°, 45° and 60° are 0, 1/√3, 1 and √3 respectively. # 1

Malika and her husband have bought a triangular corner table, which has a right angle. Given that it has a hypotenuse of 4 cm and an angle of 45°, calculate the length of the side adjacent to the angle.

cos(45°) = adjacent ÷ 4 cm
adjacent = cos(45°) × 4 cm

2√2 cm # 2

Mary is cutting a piece of cardboard into a right-angle triangle to make a birthday card. Given that she wants an angle of 65° opposite a side of 4 cm, calculate the length of the adjacent side.

tan(65°) = 4 cm ÷ adjacent
adjacent = 4 cm ÷ tan(65°)
adjacent = 1.88 cm (to 3 s.f.)

1.88 cm # 3

Two right-angled triangles are arranged as shown in the diagram. Calculate the size of the angle x.

Consider the triangle on the left.
tan(30°) = opposite ÷ 3 cm
opposite = tan(30°) × 3 cm
opposite = 1.73 cm (to 3 s. f.)

1.73 cm − 0.8 cm = 0.93 cm

Consider the triangle on the right.
sinx = opposite ÷ hypotenuse
sinx = 0.93 cm ÷ 7 cm
x = sin⁻¹(0.93 cm ÷ 7 cm)
x = 7.65° (to 3 s. f.)

7.65° # 4

Rob has a right-angled piece of card, where the angle opposite a side of 3 cm is 30°. Calculate the length of the hypotenuse of the triangle.

sinx = opposite ÷ hypotenuse
sin(30°) = 3 cm ÷ hypotenuse
hypotenuse = 3 cm ÷ sin(30°)
hypotenuse = 6 cm

6 cm # 5

Rahul is building a toy boat, as shown in the diagram. Calculate the length of the pole attached to the boat he has to use to put the sail on his boat.

sinx = opposite ÷ hypotenuse
sin(50°) = opposite ÷ 8 cm
opposite = sin(50°) × 8 cm
opposite = 6.13 cm (to 3 s. f.)

length of stick = 6.13 cm + 1 cm = 7.13 cm

7.13 cm End of page