Graphs Of Sine And Cosine
Understand the graphs of sine and cosine for Queensland Year 11 Mathematical Methods (QCAA). Both curves have amplitude one and period two pi, forming a smooth wave that repeats every full turn.
You will learn to plot the maximums, minimums and intercepts, see how a coefficient changes the amplitude, and how a constant shifts the curve up or down to change its range — the basis for modelling cycles in the QCAA course.
Every question with a fully worked solution.
- Graphs Of Sine And Cosine - Video - Graphs of sine and cosine Watch
Theory
In Year 11 Mathematical Methods (QCAA, Unit 1), the graphs of \(y=\sin t\) and \(y=\cos t\) are smooth waves that repeat every \(2\pi\) with values between \(-1\) and \(1\). This page shows their key features — amplitude, period, maximums, minimums and intercepts — and how a coefficient \(a\) or a vertical shift \(b\) changes them.
Both \(y=\sin t\) and \(y=\cos t\) are periodic with period \(2\pi\): the pattern repeats every full turn. Their amplitude is \(1\) (half the distance from the maximum \(1\) to the minimum \(-1\)), so the range is \([-1,\,1]\).
The key points occur at multiples of \(\dfrac{\pi}{2}\). The sine curve starts at \((0,0)\), peaks at \(\left(\dfrac{\pi}{2},1\right)\), and troughs at \(\left(\dfrac{3\pi}{2},-1\right)\); the cosine curve starts at its maximum \((0,1)\).
A coefficient scales the wave: \(y=a\sin t\) and \(y=a\cos t\) have amplitude \(|a|\). Adding a constant shifts it vertically: \(y=\sin t+b\) moves the whole curve up by \(b\), giving range \([b-|a|,\ b+|a|]\).
For the basic curves \(y=\sin t\) and \(y=\cos t\):
For \(y=a\sin t\) or \(y=a\cos t\):
For a vertical shift \(y=a\sin t+b\) (or with cosine):
How to read or sketch a sine/cosine graph
- Amplitude: read \(|a|\) from the coefficient, or compute \(\dfrac{\text{max}-\text{min}}{2}\) from a graph.
- Midline and range: the vertical shift \(b\) is the midline; the range is \([b-|a|,\ b+|a|]\).
- Key points: mark the maximum, minimum and intercepts at multiples of \(\dfrac{\pi}{2}\) across one period \(2\pi\).
Amplitude is half the distance from the maximum to the minimum:
| \(\text{amplitude}\) | \(=\) | \(\dfrac{1-(-1)}{2}\) |
| \(=\) | \(1\) |
The curve repeats every full turn:
| \(\text{period}\) | \(=\) | \(2\pi\) |
The maximum is \(1\) at \(t=\dfrac{\pi}{2}\) and the minimum is \(-1\) at \(t=\dfrac{3\pi}{2}\).
Amplitude \(1\), period \(2\pi\); max \(1\) at \(\dfrac{\pi}{2}\), min \(-1\) at \(\dfrac{3\pi}{2}\).
The amplitude is \(|a|\) with \(a=4\):
| \(\text{amplitude}\) | \(=\) | \(|4|\) |
| \(=\) | \(4\) |
There is no horizontal scaling, so the period is unchanged, and the range follows:
| \(\text{period}\) | \(=\) | \(2\pi\) |
| \(\text{range}\) | \(=\) | \([-4,\ 4]\) |
Cosine starts at its maximum: max \(4\) at \(t=0\) and \(t=2\pi\); min \(-4\) at \(t=\pi\).
Amplitude \(4\), period \(2\pi\), range \([-4,\,4]\); max \((0,4)\), min \((\pi,-4)\).
Amplitude is \(|a|\) with \(a=2\); the constant \(b=-1\) is the vertical shift (midline):
| \(\text{amplitude}\) | \(=\) | \(|2|=2\) |
| \(\text{midline}\) | \(=\) | \(y=-1\) |
The range runs from \(b-|a|\) to \(b+|a|\):
| \(\text{range}\) | \(=\) | \([\,-1-2,\ -1+2\,]\) |
| \(=\) | \([-3,\ 1]\) |
So the maximum value is \(1\) (at \(t=\dfrac{\pi}{2}\)) and the minimum is \(-3\) (at \(t=\dfrac{3\pi}{2}\)).
Amplitude \(2\), shift \(-1\), range \([-3,\,1]\); max \(1\), min \(-3\).
The amplitude is half the max-to-min distance:
| \(a\) | \(=\) | \(\dfrac{\text{max}-\text{min}}{2}\) |
| \(=\) | \(\dfrac{4-(-2)}{2}\) | |
| \(=\) | \(3\) |
The vertical shift is the midline, halfway between them:
| \(b\) | \(=\) | \(\dfrac{\text{max}+\text{min}}{2}\) |
| \(=\) | \(\dfrac{4+(-2)}{2}\) | |
| \(=\) | \(1\) |
A cosine starts at its maximum, so the rule is \(y=3\cos t+1\).
\(a=3,\ b=1\), so \(y=3\cos t+1\).
Common pitfalls
Frequently asked questions
What is the period of y = sin t and y = cos t?
Both have period \(2\pi\): the graph repeats every full turn.
What is the amplitude of y = a sin t?
The amplitude is \(|a|\), the absolute value of the coefficient. For example \(y=4\cos t\) has amplitude \(4\).
How do the graphs of sine and cosine differ?
They are the same shape, but shifted: \(y=\sin t\) starts at \((0,0)\) rising, while \(y=\cos t\) starts at its maximum \((0,1)\).
How does adding a constant change the graph?
\(y=\sin t+b\) shifts the whole curve up by \(b\), moving the midline to \(y=b\) and the range to \([b-|a|,\ b+|a|]\).
How do you read amplitude and vertical shift off a graph?
Amplitude \(=\dfrac{\text{max}-\text{min}}{2}\) and the vertical shift \(b=\dfrac{\text{max}+\text{min}}{2}\) (the midline).