Introduction to Trigonometry

Ratios of the sides of a right-angled triangle, their values at standard angles, the three identities, and heights and distances.

Key ideas

For an acute angle A in a right triangle: sin A = opposite ÷ hypotenuse, cos A = adjacent ÷ hypotenuse, tan A = opposite ÷ adjacent.
tan A = sin A ÷ cos A; cosec, sec and cot are the reciprocals of sin, cos and tan.
The values depend only on the angle, not on the size of the triangle.
sin A and cos A are never more than 1 for an acute angle; tan 90° is not defined.
The angle of elevation is measured up from the horizontal; the angle of depression is measured down from the horizontal.
In heights and distances, draw the right triangle first and mark the side you know and the side you want.

Standard values	sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
	cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
	tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
Identities	sin²A + cos²A = 1; 1 + tan²A = sec²A; 1 + cot²A = cosec²A

If cot θ = 20/21, evaluate ((1 + sin θ)(1 − sin θ)) ÷ ((1 + cos θ)(1 − cos θ)).

Working: = (1 − sin² θ)/(1 − cos² θ) = cos² θ/sin² θ = cot² θ = 400/441

Answer: 400/441