Overview
The shapes and key features of the sine, cosine and tangent graphs, and how transformations change them.
The graphs
- y = sin x and y = cos x — wave shape, range −1 to 1, period 360°
(2π). cos is sin shifted 90° left.
- y = tan x — period 180°, with asymptotes at 90°, 270°, …
Transformations of y = a sin(bx) + c
- a changes the amplitude (height).
- b changes the period (period = 360° ÷ b).
- c shifts the graph up/down.
Exact values worth knowing
sin 30° = ½, cos 60° = ½, sin 45° = cos 45° = 1/√2, tan 45° = 1, sin 60° = √3/2.
Worked example
y = 3 sin 2x has amplitude 3 and period 180°.
Common mistakes
- Mixing up which transformation affects amplitude (a) vs period (b).
Exam tips
- Count how many complete waves fit in 360° to read the period from a graph.