Integrated Math 3 / Unit 5: Trigonometric Functions
5.3 Graphs of Sine and Cosine
Notes
For $y = A\sin(Bx)$ or $y = A\cos(Bx)$:
- Amplitude $= |A|$. The graph oscillates between $-|A|$ and $|A|$.
- Period $\displaystyle = \frac{2\pi}{|B|}$, the length of one full cycle.
- A negative $A$ reflects the graph over the $x$-axis.
- $y = \sin x$ starts at 0 and rises. $y = \cos x$ starts at its maximum 1.
- Example: $y = 4\sin(3x)$ has amplitude 4 and period $\displaystyle \frac{2\pi}{3}$.
Practice
What are the amplitude and period of $y = 4\sin(3x)$?
Amplitude is $|A| = 4$ and period is $\displaystyle \frac{2\pi}{B} = \frac{2\pi}{3}$.
What is the period of $y = \cos(\pi x)$?
Period $\displaystyle = \frac{2\pi}{\pi} = 2$.
What is the range of $y = -2\cos x$?
The amplitude is 2. The negative sign flips the graph but does not change the range.
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