What Are the Amplitude and Period of sin x, cos x, and tan x?

The functions $y=\sin x$ and $y=\cos x$ both have amplitude $1$ and period $2\pi$. The function $y=\tan x$ has no amplitude and has period $\pi$.

Why These Values Occur

The amplitude is half the vertical distance between a periodic function's maximum and minimum values:

$$\text{amplitude}=\frac{\text{maximum}-\text{minimum}}{2}.$$

Both sine and cosine have range $[-1,1]$, so their amplitude is

$$\frac{1-(-1)}{2}=1.$$

Tangent has no maximum or minimum because it is unbounded near its vertical asymptotes. Therefore, its amplitude is not defined. A common misconception is that tangent has amplitude $1$ because $\tan 0=0$; however, amplitude depends on the function's complete vertical range, not one value.

The period is the smallest positive horizontal interval after which the graph repeats. Sine and cosine complete one cycle over $2\pi$:

$$\sin(x+2\pi)=\sin x, \qquad \cos(x+2\pi)=\cos x.$$

Tangent repeats after $\pi$:

$$\tan(x+\pi)=\tan x.$$

These periods follow from the unit circle. After a rotation of $2\pi$, the sine and cosine coordinates return to the same values. Tangent is the ratio $\frac{\sin x}{\cos x}$; both numerator and denominator change sign after $\pi$, so the ratio is unchanged. This explains why tangent repeats twice as often as sine or cosine.

For transformed functions, $y=a\sin(bx)$ and $y=a\cos(bx)$ have amplitude (|a|) and period $\frac{2\pi}{|b|}$. The function still has no amplitude, and its period is .

For example, for $y=-3\cos(2x)$, the amplitude is $|-3|=3$, while the period is $\frac{2\pi}{2}=\pi$. The negative sign reflects the graph in the -axis but does not change its amplitude. For , the vertical factor changes steepness only: amplitude remains undefined and the period is . On a graph, measure one complete repeating cycle between matching points, such as consecutive maxima or corresponding vertical asymptotes.

IB Exam Technique

This is Mathematics: Analysis and Approaches SL 3.7, examinable on Papers 1 and 2. For a graph or equation, identify the vertical scale factor and the coefficient of $x$; do not confuse the period with the horizontal stretch factor. Unless degrees are specified, IB questions assume radians.