How do you find the period of y equals a sine n t?

The period is two pi divided by n. Divide two pi by the number multiplying t. A period of pi means n equals two.

How do you find the period of y equals a sine n t?

The period is two pi divided by n. Divide two pi by the number multiplying t. A period of pi means n equals two.

Year 11 Methods (Unit 1 & 2) Trigonometric Functions

Sketch Graphs Ofy=Asinn(T±Ε) Andy=Acosn(T±Ε)

Q: What is the amplitude of a sine or cosine graph?

The amplitude is the size of a, the coefficient in front of the sine or cosine. It is the height from the centre line to a peak, and is always positive.

Q: Which way does a phase shift move the graph?

Write the argument as n times t minus e. Then t minus e moves the graph e to the right and t plus e moves it e to the left.

Q: What does a negative value of a do?

It reflects the graph in the t-axis. The amplitude is still the size of a. A maximum becomes a minimum and a minimum becomes a maximum.

Q: How do I get n from a period read off a graph?

Use n equals two pi divided by the period. Measure one full cycle, then divide two pi by that length.

20 practice questions 1 video lesson Theory + worked examples

Learn to sketch transformed sine and cosine graphs for Queensland Year 11 Mathematical Methods (QCAA). Adjusting the parameters stretches and slides the basic wave, changing its height and repeat rate.

You will explore the effect of the parameters — reading the amplitude from the leading number, finding the period from the frequency, and applying the horizontal shift — then mark key points for an accurate graph in the QCAA course.

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Practice questions

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Sketch Graphs Ofy=Asinn(T±Ε) Andy=Acosn(T±Ε) - Video - Horizontal translations of graphs of sine and cosine Watch

Theory

In Year 11 Mathematical Methods (QCAA, Unit 1), the graph of \(y=a\sin n(t-e)\) or \(y=a\cos n(t-e)\) is the basic sine or cosine wave after three changes: the amplitude \(|a|\) stretches it vertically, the factor \(n\) changes the period to \(\dfrac{2\pi}{n}\), and \(e\) is a horizontal (phase) shift. This page shows how to read those features and sketch the graph.

The amplitude is \(|a|\): the height from the centre line to a maximum. If \(a\lt 0\) the curve is reflected in the \(t\)-axis but the amplitude is still \(|a|\). The maximum value is \(|a|\) and the minimum is \(-|a|\).

The period is the horizontal length of one complete cycle, \(\dfrac{2\pi}{n}\). A larger \(n\) squeezes more cycles into the same interval.

The phase shift \(e\) slides the whole curve sideways. Written in the form \(y=a\sin n(t-e)\), the graph of \(y=a\sin nt\) moves \(e\) units to the right; a \(+e\) inside moves it \(e\) units to the left.

Factorise the inside first. Write the argument as \(n(t-e)\) so you can read \(n\) (the period) and \(e\) (the shift) directly — the shift is \(e\), not the number sitting next to \(t\).

Amplitude: \(y=3\sin t\) (navy) reaches \(\pm3\); \(y=\sin t\) (teal) only \(\pm1\).

Period: \(y=\cos 2t\) has period \(\pi\), so two cycles fit on \([0,2\pi]\).

For \(y=a\sin n(t-e)\) and \(y=a\cos n(t-e)\):

\[\text{amplitude}=|a|\]

amplitude = | a |

\[\text{period}=\dfrac{2\pi}{n}\]

period = \frac{2 π}{n}

\[\text{phase shift}=e\ \text{(right if }t-e)\]

phase shift = e

Recovering \(n\): if the period is \(P\), then \(n=\dfrac{2\pi}{P}\). For example a period of \(\pi\) gives \(n=2\).

How to sketch \(y=a\sin n(t-e)\)

Amplitude: read \(|a|\); note a reflection if \(a\lt 0\). Mark the maximum \(|a|\) and minimum \(-|a|\).
Period: compute \(\dfrac{2\pi}{n}\); this is the width of one cycle.
Shift: factor the argument as \(n(t-e)\) and translate the start of the cycle to \(t=e\).
Key points: mark the maxima, minima and \(t\)-axis intercepts across one period, then repeat.

Example 1 — Amplitude and period

State the amplitude and period of \(y=3\sin t\), and sketch one cycle.

Solution

Amplitude — the coefficient of sine:

\(\|a\|\)	\(=\)	\(\|3\|\)
	\(=\)	\(3\)

Period — here \(n=1\):

\(\text{period}\)	\(=\)	\(\dfrac{2\pi}{1}\)
	\(=\)	\(2\pi\)

Maximum \(3\) at \(t=\dfrac{\pi}{2}\); minimum \(-3\) at \(t=\dfrac{3\pi}{2}\).

Amplitude \(3\), period \(2\pi\).

amplitude = 3, period = 2 π

Example 2 — A larger frequency

State the amplitude and period of \(y=2\cos 2t\).

Solution

Amplitude:

\(\|a\|\)	\(=\)	\(\|2\|\)
	\(=\)	\(2\)

Period — \(n=2\):

\(\text{period}\)	\(=\)	\(\dfrac{2\pi}{2}\)
	\(=\)	\(\pi\)

So the curve completes two full cycles between \(0\) and \(2\pi\).

Amplitude \(2\), period \(\pi\).

amplitude = 2, period = π

Example 3 — A phase shift

Describe how to obtain \(y=4\sin\!\left(t-\dfrac{\pi}{4}\right)\) from \(y=\sin t\).

Solution

Amplitude:

\(\|a\|\)	\(=\)	\(\|4\|\)
	\(=\)	\(4\)

Period — \(n=1\):

\(\text{period}\)

\(=\)

\(\dfrac{2\pi}{1}=2\pi\)

Shift — the argument is \(\left(t-\dfrac{\pi}{4}\right)\), so \(e=\dfrac{\pi}{4}\):

\(\text{shift}\)

\(=\)

\(\dfrac{\pi}{4}\ \text{right}\)

Stretch by factor \(4\), then translate \(\dfrac{\pi}{4}\) to the right.

shift = \frac{π}{4}

Example 4 — All three changes

State the amplitude, period and phase shift of \(y=2\sin 2\!\left(t-\dfrac{\pi}{6}\right)\).

Solution

Amplitude:

\(\|a\|\)	\(=\)	\(\|2\|\)
	\(=\)	\(2\)

Period — \(n=2\):

\(\text{period}\)	\(=\)	\(\dfrac{2\pi}{2}\)
	\(=\)	\(\pi\)

Shift — already in the form \(n(t-e)\) with \(e=\dfrac{\pi}{6}\):

\(\text{shift}\)

\(=\)

\(\dfrac{\pi}{6}\ \text{right}\)

Amplitude \(2\), period \(\pi\), shift \(\dfrac{\pi}{6}\) right.

period = π, shift = \frac{π}{6}

Common pitfalls

Reading the shift before factorising. In \(y=\sin(2t-\tfrac{\pi}{3})\) the shift is not \(\tfrac{\pi}{3}\). Factor to \(2(t-\tfrac{\pi}{6})\) first; the shift is \(\tfrac{\pi}{6}\).

Confusing \(n\) with the period. \(n\) is the frequency; the period is \(\dfrac{2\pi}{n}\). A bigger \(n\) means a shorter period.

Sign of the shift. \(t-e\) shifts to the right, \(t+e\) shifts to the left. It is the opposite of the sign inside.

Negative \(a\). \(a\lt 0\) reflects the curve in the \(t\)-axis; the amplitude is still \(|a|\), never negative.

Frequently asked questions

What is the amplitude of a sine or cosine graph?

The amplitude is \(|a|\), the coefficient in front of the sine or cosine. It is the height from the centre line to a peak, and is always positive.

How do you find the period of y = a sin n t?

The period is \(\dfrac{2\pi}{n}\). Divide \(2\pi\) by the number multiplying \(t\); a period of \(\pi\) means \(n=2\).

Which way does a phase shift move the graph?

Write the argument as \(n(t-e)\). Then \(t-e\) moves the graph \(e\) to the right and \(t+e\) moves it \(e\) to the left.

What does a negative value of a do?

It reflects the graph in the \(t\)-axis. The amplitude is still \(|a|\); a maximum becomes a minimum and vice versa.

How do I get n from a period read off a graph?

Use \(n=\dfrac{2\pi}{\text{period}}\). Measure one full cycle, then divide \(2\pi\) by that length.

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Sketch Graphs Ofy=Asinn(T±Ε) Andy=Acosn(T±Ε)

📖 Theory

How to sketch \(y=a\sin n(t-e)\)

Common pitfalls

Frequently asked questions

Theory