ย 

Trigonometric functions for OCR GCSE Maths

Trigonometric functions

This page covers the following topics:

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.

Trigonometric functions

The values for some specific trigonometric ratios are useful to know. The exact values of sine for the angles 0, 30ยฐ, 45ยฐ, 60ยฐ and 90ยฐ are are 0, 1/2, 1/โˆš2, โˆš3/2 and 1 respectively. The exact values of cosine for the angles 0, 30ยฐ, 45ยฐ, 60ยฐ and 90ยฐ are are 1, โˆš3/2, 1/โˆš2, 1/2 and 1 respectively. The exact values of tangent for the angles 0, 30ยฐ, 45ยฐ and 60ยฐ are 0, 1/โˆš3, 1 and โˆš3 respectively.

Exact values of trigonometric functions in degrees

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 45ยฐ, calculate the length of the side adjacent to the angle.

cosx = adjacent รท hypotenuse
cos(45ยฐ) = adjacent รท 4 cm
adjacent = cos(45ยฐ) ร— 4 cm
adjacent = 2โˆš2 cm

2โˆš2 cm

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 45ยฐ, calculate the length of the side adjacent to the angle.

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 65ยฐ opposite a side of 4 cm, calculate the length of the adjacent side.

tanx = opposite รท adjacent
tan(65ยฐ) = 4 cm รท adjacent
adjacent = 4 cm รท tan(65ยฐ)
adjacent = 1.88 cm (to 3 s.f.)

1.88 cm

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

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(30ยฐ) = opposite รท 3 cm
opposite = tan(30ยฐ) ร— 3 cm
opposite = 1.73 cm (to 3 s. f.)

1.73 cm โˆ’ 0.8 cm = 0.93 cm

Consider the triangle on the right.
sinx = opposite รท hypotenuse
sinx = 0.93 cm รท 7 cm
x = sinโปยน(0.93 cm รท 7 cm)
x = 7.65ยฐ (to 3 s. f.)

7.65ยฐ

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

4

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

sinx = opposite รท hypotenuse
sin(30ยฐ) = 3 cm รท hypotenuse
hypotenuse = 3 cm รท sin(30ยฐ)
hypotenuse = 6 cm

6 cm

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

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 รท hypotenuse
sin(50ยฐ) = opposite รท 8 cm
opposite = sin(50ยฐ) ร— 8 cm
opposite = 6.13 cm (to 3 s. f.)

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

7.13 cm

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.

End of page

ย