Trigonometry. Trigonometry is a method of finding out an unknown angle or side in a right angled...

Trigonometry

Trigonometry is a method of finding out an unknown angle or side in a right angled triangle

Both the triangles below are similar because: The angles are the same but the

sides are different

If we measure the height and the base:

Height Base

Small triangle

Large triangle

height

base5

0.6258

100.625

16

5 cm

10 cm

8 cm

16 cm

For both trianglesheight

0.625base

This angle is in fact 320

So as long as the value of

then this angle will always be 320

height0.625

base

This is the idea behind trigonometry

If we know 2 sides then we can find the angles in the triangle

How do we know the angle is 320 ?

We can use our calculator which has been programmed to work out the angle.

We don`t have to know the height and the base it can be any 2 sides

Depending on which 2 sides are known then we use a different button on the calculator

Names are given to the 3 sides which all refer to the angle we are trying to find

The names are:

Opposite, Adjacent and Hypotenuse

Opposite means on the other side from the angle we need.

Adjacent means next to the angle we need.

Hypotenuse means the side opposite the right angle

XOpposite

Adjacent

Hypotenuse

• Identify the names of the sides of these right angled-triangles given angle k

hypotenuse

opposite

adjacent

b

b

hypotenuse

hypotenusehypotenuse

a

a

a

b

b

opposite

oppositeopposite

c

c

c

c

adjacent

adjacent

adjacenta

k

k

k

k

Opposite, Adjacent and HypotenuseIn each case label all the sides of the triangles as Opposite (O), Adjacent (A) and Hypotenuse (H) with relation to the angle marked as “X”.

X

X

X

X

X

X

X

X

X

X

x

Using the Opposite (O), Adjacent (A) and Hypotenuse (H) to work out the missing angle

The calculator has 3 buttons which are used to find the missing angle:

Sin – short for Sine

Cos – short for Cosine

Tan – short for Tangent

Deciding which button to use depends on which sides are given

• SOH CAH TOA

• Memory Aid

• Some Old Horses Sin Opposite Hypotenuse• Can Always Hear Cos Adjacent Hypotenuse• Their Owners Approaching Tan Opposite Adjacent

• Or invent one of your own

SOH CAH TOA Divide it up into three groups Place each group of three in a triangle

starting in the bottom left of each triangle

S

O

H C

A

H T

O

A

Trigonometric Ratios

TOA

CAH

SOHoppositex

hypotenusesin

oppositex

adjacenttan

adjacentx

hypotenusecos

S

O

H

C

A

H

T

O

A

What have we got and need to find?We need an angle – x.We have the Hypotenuse and Adjacent side.

Looking at the phrase, we can use C A H

10 cm

25 cm

x

Cos (x) =

Adjacent

Hypotenuse

Example 1

Hypotenuse

Adjacent

SOH CAH TOA

10 cm

25cm

x

Cos (x) =

Adjacent

Hypotenuse

Replace A and H by 10 and 25

Cos (x) = 25

10= 0.4

We now need to convert this to an angle in degrees using the Cos-1 button!!!

x = Cos –1(0.4) = 66.42oWe always find the angle using either the Cos–

1, Sin–1 or Tan–1 buttons.

Hypotenuse

Adjacent

What have we got and need to find?We need an angle – x.We have the Opposite and Adjacent side.

Looking at the phrase, we can use TOA

20 cm

15 cm

x

Tan (x) =

Opposite

Adjacent

Example 2

Opposite

Adjacent

SOH CAH TOA

Replace O and A by 15 and 20

Tan (x) =

15

20= 0.75

We now need to convert this to an angle in degrees using the Tan-1 button!!!

x = Tan –1(0.75) = 36.67o

Tan (x) =

Opposite

Adjacent

20 cm

15 cm

x

Opposite

Adjacent

We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.

What have we got and need to find?We need an angle – x.We have the Hypotenuse and Opposite side.Looking at the phrase, we can use S O H 8 cm

12 cm

x

Sin (x) = Opposite

Hypotenuse

Example 3

Hypotenuse

Opposite

SOH CAH TOA

Sin (x) = Opposite

Hypotenuse

Replace O and H by 8 and 12

Sin (x) =

8

12= 0.666

We now need to convert this to an angle in degrees using the Sin-1 button!!!

x = Sin –1(0.666) = 41.81o

8 cm12 cm

x

Hypotenuse

Opposite

We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.

Using Trigonometry to Find a Missing Side


TOA

CAH

SOHoppositex

hypotenusesin

oppositetanx=

adjacent

adjacentx

hypotenusecos

S

O

H

C

A

H

T

O

A


SOH

oppositesinx=

hypotenuse S

O

H

The triangle can also be used to find either the opposite side or the hypotenuse

opposite x hypotenusesin

oppositehypotenuse =

sinx

CAH

adjacentx

hypotenusecos C

A

H


The triangle can also be used to find either the adjacent side or the hypotenuse

adjacent x hypotenusecos

adjacenthypotenuse =

cosx

TOA

oppositex

adjacenttan

T

O

A


The triangle can also be used to find either the adjacent side or the opposite

opposite x adjacenttan

oppositeadjacent =

tanx

Example 1 SOH CAH TOA

60o

3 m

H

What have we got and need to find?

We need the Hypotenuse H

We have an angle and the Opposite O

Looking at the phrase we can use S O H

Hypotenuse = Opposite

Sin (angle)

Opposite

Hypotenuse

S

O

H

Hypotenuse = Opposite

Sin (angle)

60o

3 m

H

Replace O by 3 and (angle) by 60o

H =3

Sin (60o)Use the Sin button on your calculator to find this value

H = 8660.03

H = 3.46410…..

H = 3.46 m to 2 d.p.Opposite

Hypotenuse


40o

3 m

A


We need the Adjacent A

We have an angle and the Opposite O

Looking at the phrase we can use T O A

Opposite

Adjacent

T

O

Aopposite

adjacent = tanx

Replace O by 3 and (angle) by 40o

H =3

Tan (40o)Use the Tan button on your calculator to find this value

H = 3

0.8390

H = 3.575…..

H = 3.58 m to 2 d.p.

oppositeadjacent =

tanx

3 m

A

Opposite

Adjacent

40o


70oA

8


We need the Adjacent A

We have an angle and the Hypotenuse H

Looking at the phrase we can use C A H

Adjacent

Hypotenuse

C

A

H adjacent x hypotenusecos

70o

A

8

Replace H by 8 and (angle) by 70o

A = cos70 x 8 Use the Cos button on your calculator to find this value

H = 0.342 x 8

H = 2.736…..

H = 2.74 m to 2 d.p.

Hypotenuse

adjacent x hypotenusecos

Adjacent

Trigonometry. Trigonometry is a method of finding out an unknown angle or side in a right angled...

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