SO

HS

OH

SO

HS

OH

SO

H

CAHCAHCAHCAHCAHCAHCAHCAHCAHCAH

TOATOATOATOATOATOATOATOATOATOA

Simple Trigonometry

Moving on from Pythagoras’ Theorem, we can use Trigonometry to find a missing angle of a right

angle triangle, or the length of an unknown side of a right angle triangle, if the angle and the length of

a side are known.

90o Φ

The Formulas

In Simple Trigonometry we use three main formulas and use the acronyms SOH CAH

TOA to remember them.

(SOW – KA – TOE – WA)

SOH

Sine of the Angle = Opposite

Hypotenuse

Also shown as: -

Sin Φ = Opposite

Hypotenuse

SOH

This means the number you get when you divide the length of the opposite side of the triangle, by the length of the hypotenuse side

5

3

3

5= 0.6

This number is the SINE of the angle

CAH

Cosine of the Angle = Adjacent

Hypotenuse

Also shown as: -

Cos Φ = Adjacent

Hypotenuse

CAH

This means the number you get when you divide the length of the adjacent side of the triangle, by the length of the hypotenuse side

5

4

4

5= 0.8

This number is the COSINE of the angle

TOA

Tangent of the Angle = Opposite

Adjacent

Also shown as: -

Tan Φ = Opposite

Adjacent

TOA

This means the number you get when you divide the length of the opposite side of the triangle, by the length of the adjacent side

3

4

3

4= 0.75

This number is the TANGENT of the angle

Try transposing all three formulas

Sin Φ = Opposite

Hypotenuse

Cos Φ = Adjacent

Hypotenuse

Tan Φ = Opposite

Adjacent

The SINE, COSINE and TANGENT of the angle are not a measurement. They are a ratio made up of the lengths of two sides of the triangle.

We can use this number to find the actual angle in degrees. This used to be done using tables, but is now achieved by using a calculator

All three of the SOH CAH TOA formulas are simple divisions, but the answer gives the Sine, Cosine and

Tangent of the angle. To find the angle we need to use the inverse Sine, Cosine and Tangent function to find the actual angle. On a calculator these are shown as: -

Cos-1

Sin-1

Tan-1

These functions are accessed by pressing the SHIFT, or 2nd Function

It is helpful to recognise that the internal angles of triangle will always add up to 180 degrees and in a right angle

triangle one angle will always be 90 degrees. Using these rules we can find any missing angle, or missing side length if there is enough information in the triangle.

90o Φ

Lets try a SIMPLE example

90o Φ

4m

In this example Φ = 30o

Find the length of the Hypotenuse.

Lets try a SIMPLE example

90o Φ

4m

Using the formula Cos 30o = Adjacent

Hypotenuse

We can transpose to find: -

Hypotenuse = 4

Cos 30o

Hypotenuse = 4.62m

There are two ways to find the answer.

Using Pythagoras Theorem

Using Trigonometry Formulas

90o Φ

4m

Now find the length of the opposite side

4.62m

Pythagoras tells us: -

c = √ a2 – b2

90o Φ

4m

Now find the length of the opposite side

4.62m

c = √ (4.622 – 42)

c = 2.31m

We can also say: -

opposite = hypotenuse x Sin 30o

90o Φ

4m

Now find the length of the opposite side

4.62m

opposite = 4.62 x Sin 30o

opposite = 2.31m

Deciding which of the trigonometry formulas to use may seem complicated, but look at the problem and

see what information you have been given. If you have measurements for the opposite and hypotenuse

side, use SOH.

If you have Φ and the adjacent side you can use CAH, or TOA to find the answer.

Try This One

90o Φ

30m

a = ?m

c = ?m

If Φ = 39.64o Find the length of the hypotenuse and opposite sides.

hypotenuse = adjacent

Cos Φ

opposite = hypotenuse x Sin Φ

Try This One

90o 39.64o

30m

39.05m

25m

Based on what you have learned already what is the value of the angle marked θθ

Just One More

90o Φ

125mm

a = ?m

95mm

For this example find the length of the hypotenuse and the angle marked as Φ

a2 = b2 + c2

a2 = 1252 + 952

a = √ 1252 + 952

a = 157mm

Just One More

90o Φ

125mm

157mm

95mm

For this example find the angle marked as Φ

Cos Φ = adjacent

hypotenuse

Cos Φ = = 0.796178 125

157

Cos -1 Φ = 37.23o

Last One I Promise!!!

90o36.87o

125mm

157mm

95mm

θ

What is the value of the angle marked θ?