# Trigonometry Tangent 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 - Tangent Rule to Find the Opposite Side

A very tall Norfolk Island pine tree is in a park next to my house. Because of its age, it has root rot and is likely to fall on my house if we get a strong wind. I need to find the height of the tree to prove to the local council that it is dangerous and must be chopped down.

If I stand **20 metres** away from the base of the tree, the angle from my feet to the top of the tree is **60°**. What is the **height** of the tree?

**Answer:**

tan θ = O/A
*(Always draw a diagram and write the rule. Then substitute the numbers and letters specific to this question. See the next line of working.)*

tan 60 = x/20
*(If x is on the top of the fraction, multiply both sides of the equation by the number on the bottom which is 20.)*

tan 60 × 20 = x
*(Now type tan 30 × 20 on your calculator. You may need to close the bracket after the 30.)*

x = 34.64

The height of the tree is **34.64 metres**.

## Question - Finding the Opposite Side

If I stand **30 metres** away from the base of a building, the angle from my feet to the top of the building is **80°**. How high is the building? (Draw a diagram.)

**Answer**

170.14 metres

## Example Two - Tangent Rule to Find the Adjacent Side

At a cinema, a wheelchair ramp must be built next to steps that are **3 metres high**. For safety, the maximum angle of the ramp is **5°**.
What must be the **horizontal length** of the ramp?

**Answer:**

tan θ = O/A
*(Always draw a diagram and write the rule. Then substitute the numbers and letters specific to this question. See the next line of working.)*

tan 5 = 3/x
*(If x is on the bottom of the fraction, multiply both side by x and divide both sides by tan 5. In effect, you will swap the tan and the side. See the next line of working.)*

x = 3 / tan 5
*(Now type 3 / tan 5 on your calculator. You may need to close the bracket after the 5.)*

x = 34.29

The horizontal length of the ramp must be **34.29 metres**.

## Question - Finding the Adjacent Side

A rescue helicopter is flying at an altitude of **2000 metres** and is returning to the base.
At the base, the waiting paramedics can see that the helicopter is at an angle of **10°** from them.
What is the horizontal distance that the helicopter is away from the base? (Draw a diagram.)

**Answer**

11342.56 metres

## Example Three - Tangent Rule to Find an Angle

One section of a rollercoaster makes a right-angled triangle shape with a **vertical length of 60 metres** and a **horizontal height of 25 metres**.
What is the **angle** that the rollercoaster makes with the ground?

**Answer:**

tan θ = O/A
*(Always draw a diagram and write the rule. Then substitute the numbers and letters specific to this question. See the next line of working.)*

tan θ = 60/25

θ = tan^{-1} (60/25)
*(Now type Shift Tan (60/25) on your calculator. Use brackets for the fraction part.)*

θ = 67.38°

The angle of the rollercoaster is **67.38°**.

## Question - Finding an Angle

Another section of the rollercoaster has a **vertical height of 100 metres** and a **horizontal distance of 50 metres**. What angle does it make with the ground?

(Draw a diagram.)

**Answer**

63.43°