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Math101learn.math101.caGraphs of Sine and Cosine
Sine and cosine graphs turn circular motion into smooth repeating waves described by amplitude, period, midline, and phase.
Sine and cosine trace the vertical and horizontal coordinates of uniform motion around a circle.
Parent graphs
The parent functions are
Both have domain all real numbers, range $[-1,1]$, amplitude $1$, midline $y=0$, and period $360^\circ$ or $2\pi$ radians.
Sine begins at its midline and rises. Cosine begins at a maximum. Their shapes are identical except for a horizontal shift.
Five key points
One sine cycle in degrees passes through
One cosine cycle passes through
These points occur at quarter-period intervals. A smooth curve connects them; a sinusoid is not a series of straight segments.
Transformation form
A transformed sinusoidal function can be written
or
Its amplitude is $|a|$, its midline is $y=c$, and its phase shift is $d$. A negative $a$ reflects the wave across its midline.
Period and horizontal scale
In degrees,
In radians,
Divide the period by $4$ to space the five key points. As with other inside transformations, a larger $|k|$ compresses the graph horizontally.
Worked example: analyze a sine graph
Worked example: analyze a reflected cosine
For
the amplitude is $2$, midline $y=5$, period $360^\circ$, maximum $7$, and minimum $3$. Because $a<0$, the graph begins at a minimum rather than a maximum when $x=0$.
The range is $[5-2,5+2]=[3,7]$.
Finding an equation from a graph
Read maximum $M$ and minimum $m$ first:
Measure the horizontal distance between repeating matching points to find the period, then calculate $k$. Choose sine when a convenient starting point is a midline crossing; choose cosine when it is a maximum or minimum. Multiple equivalent equations may model the same graph.
Sine and cosine are phase shifts
In degrees,
In radians,
This identity explains why either family can describe the same sinusoidal data with a different phase shift.
Domain, range, and intercepts
For a transformed sinusoid without a contextual restriction, the domain is all real numbers and the range is
Intercepts can be found graphically or by solving a trigonometric equation. Since cycles repeat, list solutions over the requested interval rather than giving one answer without a domain.
Common mistakes
Calling $a$ the maximum. The maximum is $c+|a|$; amplitude is $|a|$.
Using $k$ as the period. Period is $360^\circ/|k|$ or $2\pi/|k|$.
Forgetting to factor the inside. Rewrite $2x-60^\circ$ as $2(x-30^\circ)$.
Spacing points by the whole period. Key points are one quarter-period apart.
Connecting key points with straight lines. The graph changes smoothly.
Quick self-check
- What are amplitude, midline, and range?
- What is the period in the correct angle unit?
- Is the phase shift read from factored form?
- Where are the five quarter-period key points?
- Does the starting direction match the sign of $a$ and the selected parent function?
Related topics
Explore the idea
Function transformation
Change one quantity at a time and connect what moves to Graphs of Sine and Cosine.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For y = 3 sin(2(x − 30°)) + 1, what are the amplitude, period, and midline?
- Amplitude = |3| = 3.
- Period = 360°/2 = 180°.
- The vertical shift gives midline y = 1.
End of lesson
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