## What is Trigonometry?

The word Trigonometry originates from the Greek words “trigonon” (triangle) and “metron” (measure). In simple terms, Trigonometry is the study of triangles, the relationship of their sides and the angles between their sides.

A right-angled triangle has 3 sides: Hypotenuse, Opposite and Adjacent. These are shown below.

Hypotenuse = Longest side

Opposite = side opposite angle

Adjacent = side adjacent to angle

## Definition of Trigonometric Ratios

COS θ = ^{A}/_{H}

SIN θ = ^{O}/_{H}

TAN θ = ^{O}/_{A}

(Cosine Ratio)

(Sine Ratio)

(Tangent Ratio)

To remember which trigonometric ratio to use in a problem, remember the following:

**SOH CAH TOA**

SOH – stands for Opposite over Hypotenuse

CAH – stands for Adjacent over Hypotenuse

TOA – stands for Opposite over Adjacent

## Finding side lengths

Trigonometry allows us to find an unknown side length of a right-angled triangle. This can be worked out by using one of the 3 ratios depending what information is given. An example is shown below:

Example 1a:

Find the length of side “a”:

Step 1: Label each side of the triangle as shown below:

Step 2: Identify appropriate trigonometric ratio. In this case it is SOH because we have the value of the Hypotenuse and we want to find the value of the Opposite side.

SIN ∅ = ^{O}/_{H}

Step 3: Substitute O=a H=6 θ=35°

SIN 35° = ^{a}/_{6}

Step 4: Now we make “a” the subject:

a = SIN 35° × 6

a = 3.441cm

## Finding Angles

We can also use Trigonometry to find unknown angles. Depending what information we are given, we can use one of the three ratios.

Example 2a:

Referring to the image below, find the value of angle “z”:

Step 1: Label the sides of the triangle:

Step 2: Substitute the values:

O=3.5cm H=5cm

SIN θ = ^{3.5}/_{5}

SIN θ = 0.7

Step 3: Make Ø the subject by using inverse sine:

θ = sin^{-1} (0.7)

θ = 44°