1. The three trigonometric ratiosθ O P S I T E H Y N U
A D J A C E N T
Sin θ =
Opposite
Hypotenuse
S O H
Cos θ =
Adjacent
Hypotenuse
C A H
Tan θ =
Opposite
Adjacent
T O A
Stress to pupils that they must learn these three trigonometric ratios.
Pupils can remember these using SOHCAHTOA or they may wish to make up their own mnemonics using these letters.
Remember:
S O H
C A H
T O A

2. Finding side lengths
If we are given one side and one acute angle in a right-angled triangle we can use one of the three trigonometric ratios to find the lengths of other sides. For example,
Find x to 2 decimal places.
We are given the hypotenuse and we want to find the length of the side opposite the angle, so we use:
56° x
12 cm
sin θ =
opposite
hypotenuse
Discuss how we decide which trigonometric ratio to use.
sin 56° = x 12
x = 12 × sin 56°
= 9.95 cm

3. What is the distance between the base of the ladder and the wall?
Finding side lengths
A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground.
What is the distance between the base of the ladder and the wall?
We are given the hypotenuse and we want to find the length of the side adjacent to the angle, so we use:
5 m
cos θ =
adjacent
hypotenuse
70° x 5 x
cos 70° =
x = 5 × cos 70°
= 1.71 m (to 2 d.p.)

4. Finding side lengths

5. The inverse of sin 30° 0.5 sin θ = 0.5, what is the value of θ?
To work this out use the sin–1 key on the calculator.
sin–1 0.5 =
30°
sin–1 is the inverse of sin. It is sometimes called arcsin.
sin
Make sure that pupils can locate the sin–1 key on their calculators.
Stress that sin and sin–1 are inverse functions.
sin 30° = 0.5 and sin–1 0.5 = 30°.
Remind pupils of the use of –1 to denote the multiplicative inverse or reciprocal. This is an extension of this notation.
sin–1
30°
0.5

6. The inverse of cos 60° 0.5 Cos θ = 0.5, what is the value of θ?
To work this out use the cos–1 key on the calculator.
cos–1 0.5 =
60°
Cos–1 is the inverse of cos. It is sometimes called arccos.
cos
Make sure that pupils can locate the cos–1 key on their calculators.
Stress that cos and cos–1 are inverse functions.
cos 60° = 0.5 and cos–1 0.5 = 60°.
cos–1
60°
0.5

7. The inverse of tan 45° 1 tan θ = 1, what is the value of θ?
To work this out use the tan–1 key on the calculator.
tan–1 1 =
45°
tan–1 is the inverse of tan. It is sometimes called arctan.
tan
Make sure that pupils can locate the tan–1 key on their calculators.
Stress that tan and tan–1 are inverse functions.
tan 45° = 1 and tan–1 1 = 45°.
tan–1
45° 1

8. Finding angles Find θ to 2 decimal places.
5 cm
8 cm
Find θ to 2 decimal places.
We are given the lengths of the sides opposite and adjacent to the angle, so we use:
tan θ =
opposite
adjacent
On the calculator we can key in tan–1 (8 ÷ 5). This avoids rounding errors when the ratio cannot be written exactly as a decimal.
tan θ = 8 5
θ = tan–1 (8 ÷ 5)
= 57.99° (to 2 d.p.)

9. Finding angles
Use this activity to practice finding the size of angles given two sides in a right-angled triangle.