Trigonometry Sine Rule

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Example One - Sine Rule to Find the Opposite Side

A mountain slope has a sloped length of 12 metres and an angle of 50° that the slope makes with level ground. What is the vertical height of the slope?

Answer:
sin θ = O/H

(Always draw a diagram and write the rule. Then substitute the numbers and letters specific to this question. See the next line of working.)

sin 50 = x/12
(If x is on the top of the fraction, multiply both sides of the equation by the number on the bottom which is 12.)

sin 50 × 12 = x
(Now type sin 50 × 12 on your calculator. You may need to close the bracket after the 50.)

x = 9.19

The vertical height of the track is 9.19 metres.

Example Two - Sine Rule to Find the Hypotenuse

A cable car at a ski resort carries skiers up to the top of the mountain. It is supported by a steel cable. The vertical height of the mountain is 3000 metres and the angle that the cable makes with the ground is 40°. What is the sloped length of the cable?

Answer:

sin θ = O/H
(Draw a diagram and write the rule. Then substitute the numbers and letters specific to this question. See the next line of working.)

sin 40 = 3000/x
(If x is on the bottom of the fraction, multiply both side by x and divide both sides by sin 40. In effect, you will swap the sin and the side. See the next line of working.)

x = 3000 / sin 40
(Type 3000 / sin 40 on your calculator. You may need to close the bracket after the 40.)

x = 4667.17 metres

The sloped length of the cable must be 4667.17 metres.

Example Three - Sine Rule to Find an Angle

For a motorbike stunt, stuntman Wild Ram Bo must ride at high speed up a ramp with a vertical height of 20 metres and a sloped length of 30 metres. What is the angle with the ground?

Answer:

sin θ = O/H
(Draw a diagram and write the rule. Then substitute the numbers and letters specific to this question. See the next line of working.)

sin θ = 20/30

θ = sin^-1 (20/30)
(Type Shift Sin (20/30) on your calculator. Use brackets for the fraction part.)

θ = 41.81°

The angle of the stunt ramp is 41.81°.