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. The values for the specific trigonometric ratios can be converted into radians by dividing angles by 180° and multiplying them by π.

The values for multiples of these angles can be found using the method of quadrants. The value of the angle stays the same, however the sign changes depending on which quadrant the angle falls in. If it falls in the first quadrant, the value is positive for all three trigonometric ratios. If it falls in the second quadrant, only sine is positive. If it falls in the third quadrant, only tangent is positive. If it falls in the fourth quadrant, only cosine is positive.

The reciprocal trigonometric ratios are given by taking the reciprocal of the usual trigonometric ratios. Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent respectively. These are usually given as cosecant, sec and cot.

Inverse trigonometric ratios are used to calculate missing angles in right-angled triangles when the value of a trigonometric ratio is known. They are given as sin⁻¹, cos⁻¹ and tan⁻¹, and are calculated using the appropriate buttons on the calculator.

Different trigonometric ratios can be related to each other by some trigonometric identities. The reciprocal trigonometric ratios can be used in identities as follows: sec²x = 1 + tan²x and cosec²x = 1 + cot²x.

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°

cos(π/4) = 1/√2
11π/4 lies in the second quadrant, therefore cosine is negative.
Therefore, cos(11π/4) = −1/√2.

−1/√2

tanx = opposite ÷ adjacent
tanx = 4.5 cm ÷ 6 cm
x = tan⁻¹(4.5 cm ÷ 6 cm)
x = 36.9° (to 3 s. f.)

36.9°

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