Sine/cosine/tangent graphs | Year 11

01 / LEARN

Understand the idea.

A cycle of sine and cosine

A sine or cosine graph records one unit-circle coordinate as the angle changes. Mark quarter-turn inputs to capture zeros and turning points, then join smoothly. Both functions repeat after $2 π$ , with outputs between $- 1$ and $1$ .

The graph of y equals sine x from minus two pi to two pi has amplitude one and period two pi. Axes retain the original pi labels, zero, one and minus one.

Tangent has asymptotes

Tangent repeats after $π$ because the coordinate quotient is unchanged by a half-turn. It is zero at $x = k π$ and undefined at $x = \frac{π}{2} + k π$ , where cosine vanishes and $k$ is an integer.

Each tangent branch covers every real output and grows without bound near its vertical asymptotes. Draw the branches separately. Since there is no finite maximum or minimum, tangent has no amplitude.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

List five key points of $y = cos x$ on $0 \leq x \leq 2 π$ .

Worked solution

1
Choose the quarter-cycle inputs.
$x = 0, \frac{π}{2}, π, \frac{3 π}{2}, 2 π$
A full cosine cycle is divided into four equal angle intervals.
2
Read the horizontal circle coordinates.
$\begin{aligned} cos 0 & = 1, & cos \frac{π}{2} & = 0, \\ cos π & = - 1, & cos \frac{3 π}{2} & = 0, \\ cos 2 π & = 1 \end{aligned}$
Cosine records horizontal position on the unit circle.
3
Pair each input with its output.
$\begin{array}{c} (0, 1), (\frac{π}{2}, 0), (π, - 1), \\ (\frac{3 π}{2}, 0), (2 π, 1) \end{array}$
The last point returns to the starting height after one complete cycle.

Answer

$(0, 1), (\frac{π}{2}, 0), (π, - 1), (\frac{3 π}{2}, 0), (2 π, 1)$

EXAMPLE 02

Worked example 2

For $y = tan x$ on $- π / 2 < x < π / 2$ , state its zero, range and asymptotes.

Worked solution

1
Find the zero of the branch.
$tan 0 = 0$
Within this interval, sine is zero only at zero and cosine there is nonzero.
2
Identify the excluded boundary lines.
$x = - \frac{π}{2}, x = \frac{π}{2}$
Cosine is zero at each boundary, so tangent is undefined and has vertical asymptotes.
3
State the output range.
$Range = ℝ$
The continuous increasing branch runs from arbitrarily negative to arbitrarily positive values.

Answer

Zero 0; range $ℝ$ ; asymptotes $x = \pm π / 2$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

What is the period of cosx?

Show answer & reasoning

$2 π$

The period is $2 π$ : after one full turn, every unit-circle coordinate repeats, so $cos (x + 2 π) = cos x$ for every real $x$ .

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

List zeros of $sin x$ on $0 \leq x \leq 2 π$ .
3 marks
Where does $cos x$ attain $- 1$ on $0 \leq x \leq 2 π$ ?
3 marks
State the range and period of $sin x$ .
3 marks
Give tangent asymptotes between $- π$ and $π$ .
3 marks
Find all $x$ with $cos x = 0$ on $- π \leq x \leq π$ .
3 marks
Compare $sin (x + 2 π)$ with $sin x$ , and $tan (x + π)$ with $tan x$ .
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

List zeros of $sin x$ on $0 \leq x \leq 2 π$ .
3 marks
Sine is zero at points on the horizontal axis.
$0, π, 2 π$
Where does $cos x$ attain $- 1$ on $0 \leq x \leq 2 π$ ?
3 marks
The leftmost unit-circle point has horizontal coordinate $- 1$ .
$x = π$
State the range and period of $sin x$ .
3 marks
Use the vertical extremes and the repeating cycle.
Range $[- 1, 1]$ ; period $2 π$ .
Give tangent asymptotes between $- π$ and $π$ .
3 marks
These are the odd half-pi values in the interval.
$x = - \frac{π}{2}, \frac{π}{2}$
Find all $x$ with $cos x = 0$ on $- π \leq x \leq π$ .
3 marks
Cosine vanishes at the vertical-axis points.
$x = \pm \frac{π}{2}$
Compare $sin (x + 2 π)$ with $sin x$ , and $tan (x + π)$ with $tan x$ .
3 marks
The added angles are whole periods of the corresponding functions.
They are equal respectively.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

In this lesson

Understand the idea.

A cycle of sine and cosine

Tangent has asymptotes

See the method in action.

Worked example 1

Choose the quarter-cycle inputs.

Read the horizontal circle coordinates.

Pair each input with its output.

Worked example 2

Find the zero of the branch.

Identify the excluded boundary lines.

State the output range.

Try it yourself.

Check your understanding

Keep practising