# Finding an Angle in a Right Angled Triangle

## Angle from Any Two Sides

We can find an **unknown angle** in a right-angled triangle, as long as we know the lengths of **two of its sides**.

### Example

The ladder leans against a wall as shown.

What is the **angle** between the ladder and the wall?

The answer is to use Sine, Cosine or Tangent!

But which one to use? We have a special phrase "SOHCAHTOA" to help us, and we use it like this:

**Step 1**: find the **names** of the two sides we know

**Adjacent**is adjacent to the angle,

**Opposite**is opposite the angle,

- and the longest side is the
**Hypotenuse**.

### Example: in our ladder example we know the length of:

- the side
**Opposite**the angle "x", which is**2.5** - the longest side, called the
**Hypotenuse**, which is**5**

**Step 2**: now use the first letters of those two sides (**O**pposite and **H**ypotenuse) and the phrase "SOHCAHTOA" to find which one of Sine, Cosine **or** Tangent to use:

SOH... |
Sine: sin(θ) = Opposite / Hypotenuse |

...CAH... |
Cosine: cos(θ) = Adjacent / Hypotenuse |

...TOA |
Tangent: tan(θ) = Opposite / Adjacent |

In our example that is** O**pposite and** H**ypotenuse, and that gives us “**SOH**cahtoa”, which tells us we need to use **Sine**.

**Step 3**: Put our values into the Sine equation:

**S**in (x) = **O**pposite / **H**ypotenuse = 2.5 / 5 = **0.5**

**Step 4**: Now solve that equation!

sin(x) = 0.5

Next (trust me for the moment) we can re-arrange that into this:

x = sin^{-1}(0.5)

And then get our calculator, key in 0.5 and use the sin^{-1} button to get the answer:

x = **30°**

### But what is the meaning of **sin**^{-1} … ?

^{-1}

Well, the Sine function * "sin"* takes an angle and gives us the

**ratio**"opposite/hypotenuse",

But * sin^{-1}* (called "inverse sine") goes the other way ...

... it takes the

**ratio**"opposite/hypotenuse" and gives us an angle.

### Example:

- Sine Function: sin(
**30°**) =**0.5** - Inverse Sine Function: sin
^{-1}(**0.5**) =**30°**

On the calculator press one of the following (depending on your brand of calculator): either '2ndF sin' or 'shift sin'. |

On your calculator, try using **sin** and **sin ^{-1}** to see what results you get!

Also try **cos** and **cos ^{-1}**. And

**tan**and

**tan**.

^{-1}Go on, have a try now.

## Step By Step

These are the four steps we need to follow:

**Step 1**Find which two sides we know – out of Opposite, Adjacent and Hypotenuse.**Step 2**Use SOHCAHTOA to decide which one of Sine, Cosine**or**Tangent to use in this question.**Step 3**For Sine calculate Opposite/Hypotenuse, for Cosine calculate Adjacent/Hypotenuse**or**for Tangent calculate Opposite/Adjacent.**Step 4**Find the angle from your calculator, using one of sin^{-1}, cos^{-1}**or**tan^{-1}

## Examples

Let’s look at a couple more examples:

### Example

Find the angle of elevation of the plane from point A on the ground.

**Step 1**The two sides we know are**O**pposite (300) and**A**djacent (400).**Step 2**SOHCAH**TOA**tells us we must use**T**angent.**Step 3**Calculate**Opposite/Adjacent**= 300/400 =**0.75****Step 4**Find the angle from your calculator using**tan**^{-1}

Tan x° = opposite/adjacent = 300/400 = 0.75

**tan ^{-1}** of 0.75 =

**36.9°**(correct to 1 decimal place)

Unless you’re told otherwise, angles are usually rounded to one place of decimals.

### Example

Find the size of angle a°

**Step 1**The two sides we know are**A**djacent (6,750) and**H**ypotenuse (8,100).**Step 2**SOH**CAH**TOA tells us we must use**C**osine.**Step 3**Calculate Adjacent / Hypotenuse = 6,750/8,100 = 0.8333**Step 4**Find the angle from your calculator using**cos**of 0.8333:^{-1}

cos a° = 6,750/8,100 = 0.8333

**cos**of 0.8333 =

^{-1}**33.6°**(to 1 decimal place)