What is the period of y = tan x?

180°. Tangent repeats every 180°.

Amplitude, period and trigonometric graphs · Cambridge IGCSE Additional Mathematics

In plain words

The graphs of sine and cosine are waves. Three numbers describe any such wave: how tall it is, how long one wave takes, and how far it has been moved up or down. All three can be read straight from the equation.

5 things to know

For y = a sin bx + c and y = a cos bx + c: the amplitude is a, the period is 360° ÷ b (or 2π ÷ b in radians), and the whole graph is moved up by c.
The greatest value is c + a and the least value is c − a. The centre line of the wave is y = c.
y = sin x starts at 0 and goes up. y = cos x starts at its highest point.
For y = a tan bx + c, the period is 180° ÷ b. A tangent graph has no amplitude. It has vertical asymptotes where bx = 90°, 270° and so on.
A larger b squeezes more waves into the same space.

Worked example

State the amplitude, period, greatest value and least value of y = 3 sin 2x + 1.

Amplitude = 3.
Period = 360° ÷ 2 = 180°.
Greatest value = 1 + 3 = 4. Least value = 1 − 3 = −2.

Worked example

State the period of y = 2 tan 3x and the equations of its asymptotes between 0° and 180°.

Period = 180° ÷ 3 = 60°.
Asymptotes where 3x = 90°, 270°, 450°: x = 30°, x = 90° and x = 150°.

Tips and tricks

The amplitude is half the distance from the lowest point to the highest. It is never negative.
On a sketch, mark the highest and lowest values on the y-axis and the ends of one period on the x-axis. For a tangent graph, draw and label the asymptotes.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Amplitude, period and trigonometric graphs: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the amplitude of y = 4 sin 3x?
- 3
- 4 (the answer)
- 12
- 120°
The number in front of sin.
What is the period of y = 4 sin 3x?
- 360°
- 90°
- 120° (the answer)
- 1080°
360° ÷ 3.
What is the greatest value of y = 2 cos x + 5?
- 2
- 5
- 7 (the answer)
- 3
5 + 2.
What is the period of y = tan x?
- 90°
- 180° (the answer)
- 360°
- 720°
Tangent repeats every 180°.
What is the least value of y = 3 sin x − 1?
- −1
- −3
- −4 (the answer)
- 2
−1 − 3.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 6 marks. Write your answers on paper, then check them.