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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 angles

We are given the lengths of the sides opposite and adjacent to the angle, so we use:

tan θ =oppositeadjacent

tan θ =85

= 57.99° (to 2 d.p.)

θ5 cm

8 cm

θ = tan–1 (8 ÷ 5)

Find θ to 2 decimal places.

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Finding angles and lengths

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Angles of depression

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The inverse of sin

To work this out use the sin–1 key on the calculator.

sin–1 0.5 = 30°

sin–1 is the inverse of sin. It is sometimes called arcsin.

30° 0.5

sin

sin–1

sin θ = 0.5, what is the value of θ?

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The inverse of cos

To work this out use the cos–1 key on the calculator.

cos–1 0.5 = 60°

cos–1 is the inverse of cos. It is sometimes called arccos.

60° 0.5

cos

cos–1

cos θ = 0.5, what is the value of θ?

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The inverse of tan

To work this out use the tan–1 key on the calculator.

tan–1 1 = 45°

tan–1 is the inverse of tan. It is sometimes called arctan.

45° 1

tan

tan–1

tan θ = 1, what is the value of θ?