The three ratios
In a right-angled triangle the ratios sine, cosine and tangent link an angle to two sides: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. These extend to any angle using the unit circle, so the ratios of angles beyond 90° follow a predictable sign pattern.
Key idea
Exact values worth memorising: sin 30° = ½, cos 60° = ½, sin 60° = √3/2, cos 30° = √3/2, tan 45° = 1, tan 60° = √3.
The graphs
The graph of y = sin x and y = cos x is a smooth wave that repeats every 360° and oscillates between −1 and 1. The graph of y = tan x repeats every 180° and has vertical asymptotes. Three features control a transformed graph:
- Amplitude: in y = a sin x the amplitude is |a| (the greatest distance from the mid-line).
- Period: in y = sin bx the period is 360°/b.
- Vertical shift: in y = sin x + c the whole curve moves up by c.
Worked example
State the amplitude, period and maximum value of y = 3 sin 2x.
Amplitude = 3. Period = 360°/2 = 180°. Maximum value = +3 (when sin 2x = 1).
Remember
- sin and cos always lie between −1 and 1.
- y = a sin bx + c has maximum a + c and minimum −a + c (for a > 0).
- The period of y = cos bx is 360°/b.