# Trigonometry Cosine 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 - Cosine Rule to Find the Adjacent Side

Kite aerial photography uses remote control to move a kite and take camera shots from a bird's eye view. The **length of the kite cable is 2000 metres**.
The **angle that the cable makes with the ground is 36°**. What is the **distance in a horizontal direction** that the kite is away from the photographer?

**Answer:**

cos θ = A/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.)*

cos 36 = x / 2000
*(If x is on the top of the fraction, multiply both sides of the equation by the number on the bottom which is 2000.)*

cos 36 × 2000 = x
*(Now type cos 36 × 2000 on your calculator. You may need to close the bracket after the 36.)*

x = 1618.03

The horizontal distance is **1618.03 metres**.

## Question - Finding the Adjacent Side

The same weather balloon is being blown by a stronger sideways breeze. The angle that the cable makes with the ground is now **40°**,
but the **cable length remains the same**. What is the **horizontal distance** from the station now? (Draw a diagram.)

**Answer**

1532.09 metres

## Example Two - Cosine Rule to Find the Hypotenuse

In Nicaragua in South America, volcano boarding is becoming a popular extreme sport. If the **horizontal distance** from the centre of the
volcanic hill is **1280 metres** and the angle of the slope with level ground is **50°**, what is the **length of the whole boarding slope**?

**Answer:**

cos θ = A/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.)*

cos 50 = 1280/x
*(If x is on the bottom of the fraction, multiply both side by x and divide both sides by cos 50. In effect, you will swap the sin and the side. See the next line of working.)*

x = 1280 / cos 50
*(Type 1280 / cos 50 on your calculator. You may need to close the bracket after the 50.)*

x = 1991.33

The sloped length is **1991.33 metres**.

## Question - Finding the Hypotenuse

After a recent volcanic eruption, there is more ash. The **new horizontal distance is 1300 metres**, but the **angle of the slope remains the same**.
How much **further** can a surfer travel from top to bottom compared with before the eruption? (Draw a diagram.)

**Answer**

31.11 metres

## Example Three - Cosine Rule to Find an Angle

A weather balloon takes measurements of altitude (height above the ground), air temperature, humidity and oxygen levels at different altitudes.
Because it contains expensive electronic devices, it is fixed by a long cable to a pole at the weather station.
The **altitude reading is 1890 metres** and the **length of cable is 2000 metres**. What is the **angle between the cable and the vertical**?

**Answer:**

cos θ = A/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.)*

cos θ = 1890 / 2000

θ = cos^{-1} (1890 / 2000)
*(Type Shift Cos (1890/2000) on your calculator. Use brackets for the fraction part.)*

θ = 19.09°

The angle is **19.09°**.

## Question - Finding an Angle

In a breeze, the weather balloon has blown further sideways. Its altitude is **1500 metres**, but the **cable is the same**.
What is the **angle** between the cable and the vertical now? (Draw a diagram.)

**Answer**

41.41°