Smooth waves between −1 and 1: y = sin x and y = cos x are smooth waves with range −1 ≤ y ≤ 1 and period 360° (2π) — they repeat every full turn. sin starts at 0 (rising); cos starts at 1.
y = sin x
- starts at (0, 0)
- max 1, min −1
- period 360° (2π)
y = cos x
- starts at (0, 1)
- max 1, min −1
- period 360° (2π)
Same wave, shifted: cos x is just sin x shifted left by 90° — same shape, different starting point.
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Repeats every 180°, with asymptotes: y = tan x is different: it has period 180° (π), no maximum or minimum (its range is all real numbers), and vertical asymptotes where cos x = 0 (at 90°, 270°, …).
Why the asymptotes?: tan x = sin x / cos x, so wherever cos x = 0 the function blows up — those are the vertical asymptotes.
No amplitude: tan has no amplitude (it isn't bounded). Amplitude only applies to sin and cos.
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Height and repeat-length: Amplitude = half the distance from max to min (how tall the wave is). Period = how far along the x-axis before it repeats. Read both straight off a graph.
IB-style question — read off a graph
A sine-type wave has maximum 5 and minimum −5, repeating every 720°. State its amplitude and period.
Step by step
- Amplitude = (max − min)/2.
- Period = repeat length.
Final answer
Amplitude 5; period 720°.
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Amplitude from max & min: Amplitude = (max − min)/2; the midline sits at (max + min)/2.
Maxes, mins, and crossings: On a basic sine wave: maximum at 90°, zero at 0°/180°/360°, minimum at 270°. Cosine: max at 0°/360°, zeros at 90°/270°, min at 180°. Knowing these anchors lets you sketch quickly.
IB-style question — where is the max?
For y = cos x on 0° ≤ x ≤ 360°, state the x-values of the maximum points.
Step by step
- cos starts at its max and returns there after a full period.
Final answer
Maxima at x = 0° and x = 360°.
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Use symmetry: Sine and cosine are symmetric — once you know one max and the period, the rest follow by spacing.