Sine, cosine and tangent are trigonometric ratios used to find angles and sides in a right-angled triangle. The sine of an angle can be calculated by dividing the opposite side to the angle by the hypotenuse. The cosine of an angle can be calculated by dividing the adjacent side to the angle by the hypotenuse. The tangent can be calculated by dividing the opposite side to the angle by the adjacent one. the three trigonometric ratios are usually given as sin, cos and tan. To find angles in a right-angled triangle, the inverse of these trigonometric ratios has to be used.

The values for some specific trigonometric ratios are useful to know. The exact values of sine for the angles 0, 30°, 45°, 60° and 90° are are 0, 1/2, 1/√2, √3/2 and 1 respectively. The exact values of cosine for the angles 0, 30°, 45°, 60° and 90° are are 1, √3/2, 1/√2, 1/2 and 1 respectively. The exact values of tangent for the angles 0, 30°, 45° and 60° are 0, 1/√3, 1 and √3 respectively.

cosx = adjacent ÷ hypotenuse
cos(45°) = adjacent ÷ 4 cm
adjacent = cos(45°) × 4 cm
adjacent = 2√2 cm

2√2 cm

tanx = opposite ÷ adjacent
tan(65°) = 4 cm ÷ adjacent
adjacent = 4 cm ÷ tan(65°)
adjacent = 1.88 cm (to 3 s.f.)

1.88 cm

Consider the triangle on the left.
tan(30°) = opposite ÷ 3 cm
opposite = tan(30°) × 3 cm
opposite = 1.73 cm (to 3 s. f.)

1.73 cm − 0.8 cm = 0.93 cm

Consider the triangle on the right.
sinx = opposite ÷ hypotenuse
sinx = 0.93 cm ÷ 7 cm
x = sin⁻¹(0.93 cm ÷ 7 cm)
x = 7.65° (to 3 s. f.)

7.65°

sinx = opposite ÷ hypotenuse
sin(30°) = 3 cm ÷ hypotenuse
hypotenuse = 3 cm ÷ sin(30°)
hypotenuse = 6 cm

6 cm

sinx = opposite ÷ hypotenuse
sin(50°) = opposite ÷ 8 cm
opposite = sin(50°) × 8 cm
opposite = 6.13 cm (to 3 s. f.)

length of stick = 6.13 cm + 1 cm = 7.13 cm

7.13 cm