Period and Amplitude of Periodic Functions

Plain $\sin\theta$ has amplitude 1 and period 360°. Two coefficients change that: one written outside the function and one inside it. In $y = A\sin(B\theta)$ they act independently — $A$ stretches the wave vertically, $B$ stretches it horizontally.

Concept A — the amplitude

\text{amplitude} = |A|

The absolute value is taken because an amplitude is always positive. A negative $A$ flips the curve upside down but leaves the amplitude at $|A|$ .

For

y = 4\cos\theta

|A| = 4

⟹ amplitude = 4, values run from −4 to 4

Concept B — the period

\text{period} = \frac{360°}{|B|} = \frac{2\pi}{|B|}

$B > 1$ speeds the function up, so it finishes a cycle in a smaller angle. $0 < B < 1$ slows it down and the cycle needs a larger angle.

Example Reading both at once

Analyse $y = 3\sin(2\theta)$ .

A = 3

, so the amplitude is

|3| = 3

B = 2

, so the period is

\dfrac{360°}{2}

⟹ amplitude = 3, period = 180° — two complete cycles in 360°

And for

y = -2\cos\!\left(\tfrac{1}{2}\theta\right)

|A| = 2

|B| = \tfrac{1}{2}

, so the period is

360° \div \tfrac{1}{2}

⟹ amplitude = 2, period = 720°, and the curve is flipped vertically

Reference The two rules

Coefficient	Where it sits	Effect
A	outside the function	amplitude = \|A\|; the period is untouched
B	inside, multiplying θ	period = 360° ÷ \|B\|; the amplitude is untouched
B > 1	—	faster: shorter period
0 < B < 1	—	slower: longer period

Summary

In y = A sin(Bθ), the amplitude is |A| — always positive.
The period is 360° ÷ |B|, or 2π ÷ |B| in radians.
A negative A reflects the curve vertically without changing the amplitude.
A and B act independently, so read them off one at a time.