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Trigonometry

The longest side in a right-angled triangle is called the hypotenuse (hyp).

The side opposite the angle is called the opposite (opp). xo The side next to the angle is called the adjacent (adj).

hypotenuse opposite xo adjacent

sin x 

opposite hypotenuse

cos x 

adjacent hypotenuse

tan x 

opposite adjacent

A useful memory aid for ing these three ratios is:

SOHCAHTOA

Finding the length of a side

Example 1 Find the value of x. x opp 24o 6 cm adj tan 24o 

opp adj

replace opp by x and adj by 6

tan 24o 

x 6

multiply both sides by 6

x  6  tan 24o x  2.67 cm (to 3 s.f.)

Example 2 Find the value of x. hyp 8 cm

cos73o 

adj hyp

replace adj by x and hyp by 8

cos 73o 

x 8

multiply both sides by 8

x  8  cos73o x  2.34 cm (to 3 s.f.)

73o x adj

Example 3 Find the value of x. hyp opp x

5 cm 40o

sin 40o 

opp hyp

replace opp by x and hyp by 5

sin 40o 

x 5

multiply both sides by 5

x  5  sin 40o x  3.21 cm (to 3 s.f.)

Example 4 Find the value of x. 15 cm adj

x hyp 65o cos 65o 

adj hyp

replace adj by 15 and hyp by x

cos 65o 

15 x

multiply both sides by x

x  cos 65o  15 x

15 cos 65o

x  35.5 cm (to 3 s.f.)

divide both sides by cos 65o

Finding an angle

Example 1 Find the value of x. opp 5 cm

hyp 9 cm xo

sin x 

opp hyp

replace opp by 5 and hyp by 9

sin x 

5 9

to find x use the sin-1 button on your calculator

�5 � x  sin-1 � � �9 � x  33.7o (to 3 s.f.)

Example 2 Find the value of x.

xo hyp 8 cm

5 cm adj

cos x 

adj hyp

replace adj by 5 and hyp by 8

cos x 

5 8

to find x use the cos-1 button on your calculator

�5 � x  cos-1 � � �8 � x  51.3o (to 3 s.f.)

Example 3 Find the value of x. 3 cm opp xo 6 cm adj tan x 

opp adj

tan x 

3 6

�3 � x  tan-1 � � �6 � x  26.6o (to 3 s.f.)

replace opp by 3 and adj by 6 to find x use the tan-1 button on your calculator

Three dimensional trigonometry

H

G F

E

5 cm D A

7 cm

B

6 cm

First you need to calculate AC using Pythagoras on triangle ABC.

To find the angle between AG and the base you need to look at triangle AGC.

G F

E

5 cm D

C A

AC 2  85

Now use trigonometry on triangle ACG. 5 tan x  85 � 5 � x  tan-1 � � � 85 �

AC  85

x  28.5o (to 3 s.f.)

AC 2  7 2  6 2

H

7 cm

B

6 cm

C

G 5 cm

x A

85

C