YEAR 9

TRIGONOMETRYWhere you see the picture below copy the information on the

slide into your bound reference.

Trigonometry is necessary in other branches of mathematics, including calculus, vectors and complex numbers.

What is Trigonometry &why study Trigonometry?

Trigonometry is the study of triangles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics.

What do you already know about right angled triangles?

The sum of the angles is 180o

One angle is equal 90o to and the sum of the other two angles is 90o

The side opposite the right angle is the longest side

Angles are written in the inside corners of

the triangle

Unknown angles are represented by symbols such as the Greek letters: Theta , Beta β, Alpha α

The little square in the corner of the triangle tells us this angle is 90o

Isosceles (90o, 45o, 45o) and scalene triangles (two angles unequal) are the two types of

right angled triangles

Unknown sides are represented

by letters in lower case.

The points, or corners, of the

triangle are labelled by letters in upper

case.

In a right angled triangle the three sides are given special names.

hypotenuse

The hypotenuse (h) is always the longest side and is opposite the right angle.

Side names of a right angled triangle

The other two sides are labelled depending on the angle you are working with.

hypotenuse

opposite

adjacent

In this case (Theta) is the angle you are working with.

The opposite (o) is the side opposite the angle .

The adjacent (a) is the side adjacent (next to) the angle .

Label the sides of these triangles

Unknown sides or angles can be found using Trigonometric ratios called sine, cosine and tangent.

Each Trigonometric ratio can be used to calculate an unknown side length or to find out an unknown angle.

First we will look into working out unknown side lengths, and we will begin by looking at Sine.

hypotenuse

opposite

adjacent

Sine is the ratio of the opposite and hypotenuse. The sine of angle . It is abbreviated to sin .

sin = opposite side length = o hypotenuse length h

opposite = sin x hypotenuse

hypotenuse

opposite

adjacent

To do this you multiply both sides by h to cancel out the divide by h. Note: h divide h = 1, and o x 1 = o. Then swap both sides of the equation.

sin = o h

sin x h = o o = sin x h

To work out the opposite side using sine you need to rearrange the formula to make the opposite (the unknown) the subject (on the left hand side of the equation).

16 mb

43o

1. Put the given values (hypotenuse and angle) into the formula: sin = o

h

sin 43o = b 16

2. Rearrange the formula to make the unknown (opposite) the subject (on the left hand side of the equation). In other words: o = sin x h.

b = sin 43o x 16

3. Calculate, and remember the units.b = 10.9m

Example for: opposite = sin x hypotenuse

hypotenuse = opposite x sin

sin = o h

sin x h = o

h = o sin

To work out the hypotenuse using sine you need to rearrange the formula to make the hypotenuse (the unknown) the subject (on the left hand side of the equation).

q19m

41o

1. Put the given values (opposite and angle) into the formula: sin = o

h

sin 41o = 19 q

2. Rearrange the formula to make the unknown (hypotenuse) the subject (on the left hand side of the equation). In other words, h = o x sin .

q = 19 sin 41o

3. Calculate, and remember the units.q = 28.96m

Example for: hypotenuse = opposite x sin

hypotenuse

opposite

adjacent

Cosine is the ratio of the adjacent and hypotenuse. The cosine of angle is abbreviated to cos .

cos = adjacent side length = a hypotenuse length h

16 m

b43o

1. Put the given values (hypotenuse and angle) into the formula: cos = a

h

cos 43o = b 16

2. Rearrange the formula to make the unknown (opposite) the subject (on the left hand side of the equation). In other words: a = cos x h.

b = cos 43o x 16

3. Calculate, and remember the units.b = 11.7m

Example for: adjacent = cos x hypotenuse

q

19m

41o

1. Put the given values (opposite and angle) into the formula: cos = a

h

cos 41o = 19 q

2. Rearrange the formula to make the unknown (hypotenuse) the subject (on the left hand side of the equation). In other words, h = a x cos .

q = 19 cos 41o

3. Calculate, and remember the units.q = 25.18m

Example for: hypotenuse = adjacent x cos

hypotenuse

opposite

adjacent

Tangent is the ratio of the opposite and adjacent. The tangent of angle is abbreviated to tan .

tan = opposite side length = o adjacent length a

16 m

b

43o

1. Put the given values (hypotenuse and angle) into the formula: tan = o

a

tan 43o = b 16

2. Rearrange the formula to make the unknown (opposite) the subject (on the left hand side of the equation). In other words: o = tan x a.

b = tan 43o x 16

3. Calculate, and remember the units.b = 14.9m

Example for: opposite = tan x adjacent

q

19m

41o

1. Put the given values (opposite and angle) into the formula: tan = o

a

tan 41o = 19 q

2. Rearrange the formula to make the unknown (hypotenuse) the subject (on the left hand side of the equation). In other words, a = o x tan .

q = 19 tan 41o

3. Calculate, and remember the units.q = 21.86m

Example for: adjacent = opposite x tan

The following mnemonic can be used to help you remember the trigonometric ratios.

SOH – CAH – TOA

The value of each of these ratios for any angle can be calculated by measuring two specific side lengths of the right-angled triangle containing that angle and dividing them.

Remembering all the trigonometric ratios

Mixed and practical problems1. Label the sides o, a, h for the

given angle.2. Use SOH – CAH –TOA to

determine which ratio to use.3. Put the values into the

formula.4. Rearrange the formula to

make the unknown the subject.

5. Calculate, remembering units.

Note: When solving practical problems – draw the diagram first.

f

8.7cm55oH

O

A

2. have a and h, so use cos

3. cos = a h

cos 55o = f 8.7

4. f = cos 55o x 8.7

5. f = 5cm

Calculating an unknown angle

To do this you need to divide by sin, cos or tan. This is called the inverse and is written and sin-1, cos-1 or tan-1.

cos = a h

= cos-1 x a h

sin = o h

= sin-1 x o h

tan = o a

= tan-1 x o a

Example for: Calculating unknown angles.

11m

16m

H

A

Ohave o and h, so use sin

sin = o h

sin = 11 16

= sin-1 11 16

= 43.43