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θ

OPPOSITE

HY

PO

TE

NU

SE

A D J A C E N T

The three trigonometric ratios

Sin θ =Opposite

Hypotenuse S O H

Cos θ =Adjacent

Hypotenuse C A H

Tan θ =OppositeAdjacent T O A

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Finding side lengths

If we are given one side and one acute angle in a right-angled triangle, we can use one of the three trigonometric ratios to find the lengths of other sides.

56°

x12 cm

To find the length of the side oppositethe angle, given the hypotenuse, use:

sin θ =opposite

hypotenuse

sin 56° =x

12

x = 12 × sin 56°

= 9.95 cm

Find x to 2 decimal places.

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Finding side lengths

A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground.

5 m

70°x

To find the length of the side adjacentto the angle, given the hypotenuse, use:

cos θ =adjacent

hypotenuse

cos 70° =x5

x = 5 × cos 70°= 1.71 m (to 2 d.p.)

What is the distance between the base of the ladder and the wall?

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Finding angles