# Trigonometry Sine Rule

Before viewing this page, it would be helpful to first read Introductory Trigonometry.

There are 3 trigonometry rules which are:

- Sin θ = O/H
- Cos θ = A/H
- Tan θ = O/A (when there is no hypotenuse)

## 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**.

## Question - Finding the Opposite Side

Another mountain has a sloped length of **32 metres** and an angle of **40°** that the slope makes with level ground.
What is the **vertical height** of the slope? (Draw a diagram.)

**Answer**

20.57 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**.

## Question - Finding the Hypotenuse

Another snowy slope has a **vertical height of 1800 metres** and the cable angle is **30°**. What is the **sloped length** of the cable? (Draw a diagram.)

**Answer**

3600 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°**.

## Question - Finding an Angle

As for the bad news, the stunt didn't work. The good news is that the Ram is recovering in hospital. The substitute rider, Wilder Spin,
constructed a new ramp with a **vertical height of 25 metres** and the same sloped length of 30 metres. What is the **angle with the ground**? (Draw a diagram.)

**Answer**

56.44°

## Did You Know That...?

Robbie Maddison of the **Crusty Demons motorbike stunt team** jumped London's Tower Bridge with a back flip. The opened drawbridge was the ramp.