Graphs of Sine and Cosine

Both $y=\sin x$ and $y=\cos x$ are periodic with period $2\pi$: the graph repeats every $2\pi$ units. Each has amplitude $1$, midline $y=0$, maximum value $1$, and minimum value $-1$.

• $y=\sin x$ starts at the midline: $(0,0)$, rises to a max, returns, falls to a min, returns.
• $y=\cos x$ starts at a maximum: $(0,1)$, falls to the midline, to a min, and back up.

Five-point method: divide one period into four equal pieces. For $[0,2\pi]$ the key $x$-values are

Evaluate the function at these five points, then connect with a smooth wave.

The number $a$ stretches the graph vertically.

• Amplitude $=|a|=\dfrac{\text{max}-\text{min}}{2}$: the distance from the midline to a peak.
• Maximum value $=|a|$, minimum value $=-|a|$ (when midline is $y=0$).
• If $a<0$, the graph is reflected across the $x$-axis (flipped upside down).
• $a$ does not change the period or the midline.

In $y=a\sin(bx)$ or $y=a\cos(bx)$ (with $b>0$), the coefficient $b$ controls how fast the wave cycles:

• $b>1$ compresses the graph: more cycles, shorter period.
• $0<b<1$ stretches the graph: fewer cycles, longer period.

To find five key points, split one period into four equal steps of width $\tfrac{1}{4}\cdot\tfrac{2\pi}{b}=\tfrac{\pi}{2b}$.

Two more numbers slide the whole graph:

• Vertical shift $d$: raises ($d>0$) or lowers ($d<0$) the graph. The new midline is $y=d$; max $=d+|a|$, min $=d-|a|$.
• Phase (horizontal) shift $=\dfrac{c}{b}$: found by setting the inside equal to zero, $bx-c=0\Rightarrow x=\dfrac{c}{b}$. A positive result shifts right, negative shifts left.

• Amplitude $=|a|$.   2. Period $=\dfrac{2\pi}{b}$.   3. Phase shift $=\dfrac{c}{b}$ (right if $+$).
• [4.] Midline $y=d$; max $=d+|a|$, min $=d-|a|$.
• [5.] Start the cycle at $x=\dfrac{c}{b}$, then plot five points a quarter-period $\dfrac{2\pi}{4b}$ apart.

Given a graph or a word description, recover $a,b,c,d$:

• $d=\dfrac{\text{max}+\text{min}}{2}$ (midline).   2. $|a|=\dfrac{\text{max}-\text{min}}{2}$ (amplitude).
• [3.] $b=\dfrac{2\pi}{\text{Period}}$.
• [4.] Choose sine or cosine to match the starting behavior, then set $c$ from the horizontal shift. A maximum at $x=x_0$ suggests $y=a\cos\!\big(b(x-x_0)\big)+d$. A midline point rising suggests sine.

A sum of a sine and a cosine of the same frequency is itself a single sinusoid. For any constants $a,b$,

• $R=\sqrt{a^2+b^2}$ is the amplitude of the combined wave (always $\ge 0$).
• The phase angle $\phi$ satisfies $\cos\phi=\dfrac{a}{R}$ and $\sin\phi=\dfrac{b}{R}$; use both signs to place $\phi$ in the correct quadrant — do not trust $\arctan$ alone.
• You may also write it as $R\cos(x-\psi)$ with $\tan\psi=\dfrac{a}{b}$. Same curve, different bookkeeping.

This is why superposition of two oscillations (of equal period) never produces a new shape — only a shifted, rescaled sine.

Once a sinusoid is written as $y=A\sin(bx-c)+d$ (or found via $R$ above), reading its extremes is mechanical:

• Maximum $=d+|A|$, reached when the inside angle $=\tfrac{\pi}{2}$; minimum $=d-|A|$, when the inside angle $=-\tfrac{\pi}{2}$ (or $\tfrac{3\pi}{2}$).
• Solve for the location: set $bx-c=\tfrac{\pi}{2}+2\pi k$ and solve for $x$ to find where the max occurs.

Period of a sum of different frequencies. $y=\sin(b_1 x)+\sin(b_2 x)$ is generally not a simple sinusoid. Its period is the least common multiple of the two periods $\tfrac{2\pi}{b_1}$ and $\tfrac{2\pi}{b_2}$ (when that LCM exists):

If the ratio $b_1/b_2$ is irrational, the sum is not periodic at all.

To find where two graphs meet, or how many times an equation is satisfied on an interval, think geometrically:

• Intersection of two curves $y=f(x)$ and $y=g(x)$: set $f(x)=g(x)$ and solve. Each solution is one crossing point.
• Number of solutions of $\sin x=k$ on $[0,2\pi)$: draw the horizontal line $y=k$. If $-1<k<1$ there are 2 solutions; if $k=\pm 1$ there is exactly 1 (a tangent touch at the peak/trough); if $|k|>1$ there are 0.
• For a compressed wave $\sin(bx)=k$ on $[0,2\pi)$, the line crosses roughly $2b$ times because the wave completes $b$ full cycles. Count crossings, do not just solve once.

From scattered data. You do not need a clean starting point. Given a few facts:

• Get the midline and amplitude from any known high and low: $d=\tfrac12(\text{high}+\text{low})$, $|A|=\tfrac12(\text{high}-\text{low})$.
• Get $b$ from the period, or from the distance between a max and the next min (that gap is a half-period).
• Pin the horizontal shift with one labeled point — ideally a max, min, or midline crossing — by forcing the model to pass through it.

Different formulas, identical graphs. Because sine and cosine are shifts of each other, many equations describe the same curve. All four below are one graph:

A reflection $-\!\sin x$ equals a half-period shift $\sin(x-\pi)$; adding $2\pi$ to a phase changes nothing. When two answers look different, test a few points (or compare $A,b,d$ and one peak location) to confirm they match.