Trigonometric functions (Grade 10) — Mathematics Grades 8–11

Sketch and interpret the graphs of sin, cos and tan from 0° to 360°.

Gr 10 Term 2 - Trigonometric functionsTerm 2 - 1 week (CAPS)

Practise Trigonometric functions in the app →

What gets asked

Complete a table of values for sin, cos or tan from 0° to 360°.
Sketch the graph of a given trig function over 0° to 360°.
State the domain, range or asymptotes of a trig graph.
Find the equation of a trig graph from its sketch.

You must be able to

Read off values of sin, cos and tan at multiples of 30° or 45°.
Mark the asymptotes on a tangent graph at 90° and 270°.
Describe a as a vertical stretch, and q as a vertical shift.
Read the amplitude or the vertical shift straight off a sketched graph.

Traps that cost marks

Writing a number for tan 90°, instead of marking it as undefined. tan 90° is undefined. Leave a gap in the table and draw an asymptote there.
Drawing the tangent graph as one unbroken curve through 90° and 270°. The tangent graph breaks at 90° and 270°. Draw two separate branches, not one.
Saying that a changes how often the graph repeats, instead of how tall it is. a stretches the graph vertically. It does not change how often the wave repeats.

Worked example

Sketch

y = 2 sin θ

for

0° \leq θ \leq 360°

. State the range.

$y = sin θ$ has range $- 1 \leq y \leq 1$ .
Multiplying by 2 stretches it, so the range becomes $- 2 \leq y \leq 2$ .
The graph still starts at $(0°; 0)$ and completes one wave by $360°$ .

Trigonometric graphs

Sine and cosine roll as waves, tangent breaks into pieces, and two numbers reshape all three.

domain: All the input values you may put into the function.
range: All the $y$ -values the graph actually reaches.
amplitude: Half the distance from the top of a wave to its bottom.
period: How far along before the graph starts repeating itself.
asymptote: A line these graphs creep closer to, but never touch.
intercept: A point where the graph cuts one of the axes.

Build these graphs point by point. Take $θ$ at $0^{\circ}$ , then $3 0^{\circ}$ , $6 0^{\circ}$ , $9 0^{\circ}$ and on to $36 0^{\circ}$ . Work out $y$ at each and plot it. The domain is $0^{\circ}$ to $36 0^{\circ}$ , the stretch you were asked to draw.

The sine graph leaves the origin, climbs to $1$ at $9 0^{\circ}$ , crosses $0$ at $18 0^{\circ}$ , dips to $- 1$ at $27 0^{\circ}$ and returns. Its range is $- 1 \leq y \leq 1$ . Cosine is the same wave started at $1$ , with its intercepts on the $θ$ -axis at $9 0^{\circ}$ and $27 0^{\circ}$ .

The tangent graph is not a wave. It has no maximum and no minimum, so do not hunt for one. It is undefined at $9 0^{\circ}$ and $27 0^{\circ}$ , with an asymptote at each. The graph breaks there, so draw it in separate pieces.

In $y = a sin θ$ the number $a$ stretches the graph up and down. For positive $a$ the amplitude becomes $a$ , and the range opens to $- a \leq y \leq a$ . A negative $a$ turns the wave upside down. What $a$ never touches is the period.

In $y = sin θ + q$ the number $q$ slides the graph up or down, never sideways. The range slides with it, to $q - 1 \leq y \leq q + 1$ . Reading an equation off a graph reverses this. Then $q$ sits halfway between maximum and minimum, and $a$ is half the gap.

Rules to remember

$y = sin θ$ has range $- 1 \leq y \leq 1$
for $a > 0$ , $y = a sin θ + q$ has range $q - a \leq y \leq q + a$
$tan 9 0^{\circ}$ and $tan 27 0^{\circ}$ are undefined

Examples

Complete a table for

y = cos θ

0^{\circ}

9 0^{\circ}

18 0^{\circ}

27 0^{\circ}

and

36 0^{\circ}

Worked answer

$cos 0^{\circ} = 1$ and $cos 9 0^{\circ} = 0$ .
$cos 18 0^{\circ} = - 1$ , the lowest it goes.
$cos 27 0^{\circ} = 0$ and $cos 36 0^{\circ} = 1$ .
So the five values run $1$ , $0$ , $- 1$ , $0$ , $1$ .

Answer: $1$ , $0$ , $- 1$ , $0$ , $1$

The wave started at the top and came back after one full turn.

For

y = 2 cos θ - 1

0^{\circ} \leq θ \leq 36 0^{\circ}

, give the range and the maximum.

Worked answer

Start from $y = cos θ$ , whose range is $- 1 \leq y \leq 1$ .
The $2$ stretches it, so the range becomes $- 2 \leq y \leq 2$ .
Taking off $1$ slides it down, giving $- 3 \leq y \leq 1$ .
The maximum is $1$ , reached at $θ = 0^{\circ}$ .

Answer: range $- 3 \leq y \leq 1$ , maximum $1$

The stretch came first, and the slide moved the whole range.

A sine graph has a maximum of

5

and a minimum of

1

. Write its equation as

y = a sin θ + q

Worked answer

The middle line lies halfway between them: $q = \frac{5 + 1}{2} = 3$ .
The stretch is half the gap: $a = \frac{5 - 1}{2} = 2$ .
So the equation is $y = 2 sin θ + 3$ .
Check: the top is $2 + 3 = 5$ and the bottom is $- 2 + 3 = 1$ .

Answer: $y = 2 sin θ + 3$

Finding the middle line first made the stretch easy to see.

Traps

Writing a number for $tan 9 0^{\circ}$ , instead of marking it as undefined. $tan 9 0^{\circ}$ is undefined. Leave a gap in the table and draw an asymptote there.
Saying that $a$ changes how often the graph repeats, instead of how tall it is. $a$ stretches the graph up and down. It does not change how often the wave repeats.
Sliding the graph with $q$ but leaving the range unchanged. The range moves too. Add $q$ to both ends of it.
Giving a maximum and a minimum for the tangent graph. It has neither. It climbs without stopping near each asymptote.

Learn and practise “Trigonometric graphs” in the app →

Grade 10 Trigonometric functions

What gets asked

You must be able to

Traps that cost marks

Worked example

Trigonometric graphs

Rules to remember

Examples

Traps

About this material