Sine/cosine/tangent graphs | Year 11

01 / LEARN

The idea, step by step

A cycle of sine and cosine

Sine starts at 0, rises to 1, returns to 0, falls to−1 and returns to 0 over one cycle. Cosine starts at 1. Both have range[−1,1] and period 2π. Use labelled quarter-cycle points, then a smooth curve, rather than a collection of arbitrary calculator points.

Tangent has asymptotes

Tangent has period π, zeros at $x = k π$ , and vertical asymptotes at $x = π / 2 + k π$ , where k is an integer. Its branches are unbounded, so it has no amplitude. An asymptote is not a set of curve points; never join a branch across it.

A common trap

Calling 1 the amplitude of tangent because tan(π/4)=1.

Instead: Tangent has no finite maximum or minimum; amplitude is not defined.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

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

Use the angles 0,π/2,π,3π/2,2π.
Read their horizontal unit-circle coordinates.
The last value repeats the first after a full cycle.

(0, 1), (π / 2, 0), (π, - 1), (3 π / 2, 0), (2 π, 1)

Example 2

Worked example 2

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

$tan 0 = 0$ .
The branch covers all real vertical values.
The two excluded boundary values are vertical asymptotes.

Zero 0; range

ℝ

; asymptotes

x = \pm π / 2

Check your understanding

What is the period of cosx?

Show answer

$2 π$

The unit-circle coordinates repeat after a full turn.

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

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

Read the worked solutions

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

List zeros of sinx on $0 \leq x \leq 2 π$ .
3 marks
Sine is zero at points on the horizontal axis.
$0, π, 2 π$
Where does cosx 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 sinx.
3 marks
Use the vertical extremes and the repeating cycle.
Range $[- 1, 1]$ ; period $2 π$ .
Give tangent asymptotes between−π andπ.
3 marks
These are the odd half-pi values in the interval.
$x = - π / 2, π / 2$
Find all x with cosx=0 on $- π \leq x \leq π$ .
3 marks
Cosine vanishes at the vertical-axis points.
$x = \pm π / 2$
Compare sin(x+2π) with sinx, and tan(x+π) with tanx.
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 ↗

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