8.4 Graphs of Sine and Cosine

Notes

The graphs of $y = \sin x$ and $y = \cos x$ are waves that repeat every $2\pi$.

• General form: $y = a\sin(b(x - h)) + k$ or $y = a\cos(b(x - h)) + k$.
• Amplitude $= |a|$ (half the distance from max to min).
• Period $\displaystyle = \frac{2\pi}{b}$ (length of one cycle).
• Midline: $y = k$. Maximum $= k + |a|$, minimum $= k - |a|$.
• $y = \sin x$ starts at the midline going up; $y = \cos x$ starts at a maximum.
• Example: $y = 3\sin(2x) + 1$ has amplitude 3, period $\pi$, midline $y = 1$, max 4, min $-2$.