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Learn · Mathematics · Grade 11 · Trigonometry

Graphs of sine, cosine and tangent

Period, amplitude and shift in y = a·sin(bx + c) + d, the tangent’s asymptotes, and radians on the x-axis.

High school Lesson 26 of 31 Interactive figure Practice

First, the rules

Unroll the unit circle

Walk around the unit circle and plot the height against the angle: that curve is the sine wave. Cosine is the same wave started a quarter turn earlier; tangent runs off to infinity where cosine is zero.

★ The basic curves
Example 1
sin x and cos x: between −1 and 1, period 360° (2π). tan x: every real value, period 180° (π), vertical asymptotes at 90° + 180°k.
Key points of sin x
Example 1
0° → 0, 90° → 1, 180° → 0, 270° → −1, 360° → 0
cos x is the same list shifted: 1, 0, −1, 0, 1.
Even and odd
cos(−x) = cos x (mirror in the y-axis) · sin(−x) = −sin x, tan(−x) = −tan x (symmetric through the origin)
★ The transformed wave
Examples 2, 3
y = a · sin(b(x − c)) + d
amplitude |a| · period 360° / b · phase shift c (right if positive) · midline y = d · max d + |a|, min d − |a|.
★ Reading a from a graph
Example 3
a = max − min2, d = max + min2
Radians
180° = π — on a radian axis the period of sin x is 2π and the asymptotes of tan x sit at π/2 + kπ.

See it

Stretch, squeeze and lift the wave

The dashed curve is the plain function. Change a to stretch it vertically (negative flips it), b to change how many waves fit in 360°, d to lift it. Switch to tangent to see the asymptotes every 180° / b.

Step by step

Worked examples

1

Sketching one period

key points

Sketch y = cos x for 0° ≤ x ≤ 360° and read off where it is negative.

Ask yourself: Plot the five key points first, then join them with a smooth wave.

Show the solution
  1. Key points: (0°, 1), (90°, 0), (180°, −1), (270°, 0), (360°, 1).
  2. The curve starts at its top, dips through 0 at 90°, bottoms out at 180°, returns to 0 at 270° and to the top at 360°.
  3. It is below the axis strictly between 90° and 270° — exactly the quadrants where cosine is negative on the unit circle.

Answer cos x < 0 for 90° < x < 270°

2

Reading the equation

amplitude · period · shift

For y = 3 sin(2x) − 1 give the amplitude, period, maximum, minimum and the first positive x where the maximum is reached.

Ask yourself: Amplitude from a, period from b, the extremes from d ± |a|.

Show the solution
  1. Amplitude 3; period 360° / 2 = 180°.
  2. Maximum −1 + 3 = 2, minimum −1 − 3 = −4.
  3. sin(2x) = 1 first when 2x = 90°, i.e. x = 45°.

Answer 3, 180°, max 2, min −4, first max at 45°

3

From a graph to an equation

reverse

A wave of the form y = a cos(bx) + d (a > 0) has maximum 5, minimum −1 and completes one full wave between x = 0° and x = 120°. Find a, b and d.

Ask yourself: Max and min give a and d; one wave in 120° gives the period, hence b.

Show the solution
  1. a = (5 − (−1)) / 2 = 3, d = (5 + (−1)) / 2 = 2.
  2. Period 120° = 360° / b, so b = 3.
  3. y = 3 cos(3x) + 2. Check: at x = 0 the cosine is 1 and y = 5, the maximum. ✓

Answer y = 3 cos(3x) + 2

Practice: read the equation

0 / 0

Amplitude, period, maximum value — or pick the function with a given period. Your best score is kept in this browser only.

About this lesson

Graphs of sine, cosine and tangent: the shape of each curve over a full turn, period, amplitude, vertical shift and phase in y = a·sin(b(x − c)) + d, the asymptotes of tangent, and radians on the axis. Move the sliders, then practise reading period and amplitude from an equation.

Part of Learn mathematics · Grade 11 · Trigonometry. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.