## Trigonometry: Right-angled triangles Quesitons

For each set of questions find the value of

Make sure to write down all the working. It is a good habit to be rigorous in your answers.

Set 1

Set 2

Set 3

Set 4

Set 5

## Trigonometry and Right-angled triangles

#### Conceptual Understanding

If we find the ratio between a pair of sides on a triangle the value will be a constant even on larger triangles provided the triangles are similar to each other.

### The right-angled trigonometry triangle

The hypotenuse is the longest side and is opposite the right-angle

The opposite is the side that is opposite the given angle. (Or the angle we wish to find)

The adjacent is the side next to the given angle. (It is between the given angle and the right angle)

### The ratios SohCahToa

When solving a trigonometry right-angled triangle problem we need two pieces of information and then use the relevant ratio to find the third.

There are three possible scenarios.

- We are finding an angle
- We are finding a side that is a numerator in the ratio
- We are finding a side that is a denominator in the ratio

## Examples

Type 1 (The unknown is the numerator)

We have the hypotenuse and we want the opposite side. The only ratio with hypotenuse and opposite in it is the sine ratio.

Let’s fill out the equation

Type 2 (The unknown is the denominator)

We have the opposite and we want the adjacent side. The only ratio with adjacent and opposite in it is the tangent ratio.

Let’s fill out the equation

Type 3 (The angle is unknown)

We have the hypotenuse and adjacent side. The only ratio with adjacent and hypotenuse in it is the cosine ratio.

Let’s fill out the equation

#### Questions

## Pythagoras’ Theorem Questions

Set 1: Find the value of a, b, c and d

Set 2: Find the value of a, b, c and d

Set 3: Find the value of a and c in the first diagram and the area of the second diagram

## Pythagoras’ Theorem

Definition: The square of the hypotenuse is equal to the sum of the squares of the other two sides.

#### Proof

What this means is that if we draw three squares where the length of the sides of the squares are equal to the lengths of the sides of the triangle the area of the big square equals the area of the two smaller squares added together. (note: The hypotenuse is opposite the right angle and this side produces the largest square)

In the diagram above the largest square is the green square so this implies that

It follows that if we are finding a shorter side then we could use

Examples

Find the length of the hypotenuse.

Here

So using

We get

Find the length of the missing sideSince we are finding one of the shorter sides we will use

Here

We get

Find the value of This is a right-angled triangle so let’s use

Does 11cm, 59cm, 60cm produce a right-angled triangle?Use where since that is the longest length.But , therefore this is not a right-angled triangle.