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Trigonometric Graphs

Trigonometric Graphs

Graphs of sin, cos and tan

Sketching Sin Cos Tan Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. The three graphs
  4. Two angles, one value
  5. Exact values
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus E6.3 and E6.4: trigonometric graphs and exact values
  • E6.4Recognise, sketch and interpret \(y=\sin x\), \(y=\cos x\) and \(y=\tan x\) for \(0^\circ\le x\le360^\circ\).
  • E6.4Use the symmetry of each graph to find a second angle with the same value.
  • E6.3Know the exact values of sin and cos for 0°, 30°, 45°, 60°, 90° and tan for 0°, 30°, 45°, 60°.

Before you start: sin, cos and tan in right-angled triangles. Next: Solving trig equations.

Formula bank

Same sineLearn it\(\sin x=\sin(180^\circ-x)\)Use it when a second angle with the same sine.
Same cosineLearn it\(\cos x=\cos(360^\circ-x)\)Use it when a second angle with the same cosine.
Same tangentLearn it\(\tan x=\tan(x+180^\circ)\)tan repeats every 180°.Use it when a second angle with the same tangent.
Exact valuesLearn it\(\sin30^\circ=\tfrac12,\ \sin45^\circ=\tfrac{\sqrt2}2,\ \sin60^\circ=\tfrac{\sqrt3}2\); \(\tan45^\circ=1,\ \tan60^\circ=\sqrt3\)cos runs the sin list backwards.Use it when 'exact value' (no calculator).

The three graphs

Picture a point going round a circle of radius 1: its height is \(\sin x\) and its distance across is \(\cos x\). That is why both stay between −1 and 1 and repeat every 360°. \(\tan x=\dfrac{\sin x}{\cos x}\) breaks where \(\cos x=0\) (90° and 270°).

Starts at 0, up first

y = sin x90180270360-1.5-1.0-0.50.51.01.5xy(90, 1)(270, −1)180
0 → 1 → 0 → −1 → 0.
FeatureValue
crosses x-axis0°, 180°, 360°
maximum1 at 90°
minimum−1 at 270°

Starts at 1, down first

y = cos x90180270360-1.5-1.0-0.50.51.01.5xy(0, 1)(180, −1)(360, 1)
1 → 0 → −1 → 0 → 1.
FeatureValue
crosses x-axis90°, 270°
maximum1 at 0° and 360°
minimum−1 at 180°

Asymptotes at 90° and 270°

y = tan x90180270360-4-3-2-11234xy(45, 1)(225, 1)
Repeats every 180°.
FeatureValue
crosses x-axis0°, 180°, 360°
asymptotesx = 90°, x = 270°
max / minnone

General methodHow to sketch y = sin x, cos x or tan x

  1. Mark 90°, 180°, 270°, 360° on the x-axis and 1, −1 on the y-axis.
  2. Plot the five key points (sin: 0, 1, 0, −1, 0; cos: 1, 0, −1, 0, 1).
  3. tan: dashed asymptotes at 90° and 270°; through 0°, 180°, 360°; (45°, 1) and (225°, 1).
  4. Join with smooth curves (waves for sin and cos, never pointed).

Try it yourself

1
Where does \(y=\cos x\) cross the x-axis (0° to 360°)? Type both angles.
Show answer

Answer 90, 270

2
Maximum value of \(y=\sin x\)?
Show answer

Answer 1

Two angles, one value

A horizontal line usually cuts the graph twice between 0° and 360°, so two angles share the same value. The symmetry of each graph gives the second angle.

sin 40° = sin 140°90180270360-1.5-1.0-0.50.51.01.5xy40°140°
The sin graph is symmetrical about x = 90°.
FunctionSame value atExample
sin\(x\) and \(180^\circ-x\)sin 40° = sin 140°
cos\(x\) and \(360^\circ-x\)cos 60° = cos 300°
tan\(x\) and \(x+180^\circ\)tan 45° = tan 225°

Try it yourself

1
Another angle (0° to 360°) with the same sine as 20°?
Show answer

Answer 160

2
Another angle with the same cosine as 110°?
Show answer

Answer 250

3
Another angle with the same tangent as 130°?
Show answer

Answer 310

Exact values

Two triangles give every exact value: half a square (45°, 45°, sides 1, 1, \(\sqrt2\)) and half an equilateral triangle (30°, 60°, sides 1, \(\sqrt3\), 2).

0°30°45°60°90°
sin0\(\tfrac12\)\(\tfrac{\sqrt2}{2}\)\(\tfrac{\sqrt3}{2}\)1
cos1\(\tfrac{\sqrt3}{2}\)\(\tfrac{\sqrt2}{2}\)\(\tfrac12\)0
tan0\(\tfrac{1}{\sqrt3}\)1\(\sqrt3\)—
Read the sin row as \(\tfrac{\sqrt0}{2},\tfrac{\sqrt1}{2},\tfrac{\sqrt2}{2},\tfrac{\sqrt3}{2},\tfrac{\sqrt4}{2}\). The cos row is the same list backwards.

Try it yourself

1
Exact value of \(\sin30^\circ+\cos60^\circ\)?
Show answer

Answer 1

2
Exact value of \(\tan45^\circ\)?
Show answer

Answer 1

Ways they test it

Type 1 Sketch

"Sketch \(y=\cos x\) for \(0^\circ\le x\le360^\circ\)." (2 marks)

Spot it a blank set of axes.

  1. Five key points.
  2. Smooth wave.

Type 2 Second angle

"\(\sin40^\circ=0.643\). Find another angle between 0° and 360° with sine 0.643." (1 mark)

Spot it one angle is given.

  1. Use the symmetry rule for that function.

For example \(180^\circ-40^\circ\)

    Answer 140°

    Type 3 Exact value

    "Write down the exact value of \(\tan60^\circ\)." (1 mark)

    Spot it 'exact'.

    1. From the table (no decimals).

    For example \(\tan60^\circ\)

      Answer \(\sqrt3\)

      Mistakes to avoid

      • ✗ sin and cos graphs swapped✓ sin starts at 0, cos at 1sin 0° = 0, cos 0° = 1.
      • ✗ tan drawn with peaks at ±1✓ tan goes off to ±∞ at 90° and 270°Asymptotes.
      • ✗ sin 40° = sin 320°✓ sin 40° = sin 140°sin uses 180° − x; 360° − x is for cos.

      Summary

      • sin: 0, 1, 0, −1, 0. cos: 1, 0, −1, 0, 1 (at 0°, 90°, 180°, 270°, 360°).
      • tan: asymptotes at 90° and 270°; repeats every 180°.
      • Same value: sin x = sin(180° − x); cos x = cos(360° − x); tan x = tan(x + 180°).
      • Exact values come from the 45° and 30°-60° triangles.

      If you see …

      • "another angle with the same …"use that function's symmetry rule.
      • "exact value"fractions and surds from the table.

      Practice

      Practice: easy, medium, hard

      Easy

      1
      Minimum value of \(y=\cos x\)?
      Show answer

      Answer −1

      2
      Where is the minimum of \(y=\sin x\) (0° to 360°)?
      Show answer

      Answer 270

      3
      Another angle with the same sine as 30°?
      Show answer

      Answer 150

      Medium

      1
      Another angle with the same cosine as 40° (0° to 360°)?
      Guide me

      What is it asking? an angle

      What do we know? cos rule

      Plan 360° − 40°

      Show working
      1. 320°

      Answer 320

      2
      \(\sin40^\circ=0.643\). Find both angles between 0° and 360° with sine −0.643.
      Guide me

      What is it asking? two angles

      What do we know? sin is negative from 180° to 360°

      Plan 180° + 40° or 360° − 40°

      Show working
      1. 220° and 320°

      Answer 220, 320

      3
      Exact value of \((\cos30^\circ)^{2}\)?
      Guide me

      What is it asking? a fraction

      What do we know? \(\cos30^\circ=\tfrac{\sqrt3}2\)

      Plan square it

      Show working
      1. \(\tfrac34\)

      Answer 3/4

      Hard

      1
      Exact value of \(\tan60^\circ\times\tan30^\circ\)?
      Guide me

      What is it asking? a number

      What do we know? \(\sqrt3\times\tfrac1{\sqrt3}\)

      Plan multiply

      Show working
      1. 1

      Answer 1

      PartAnswer / workingMarksGuidance
      1B1
      Total: 1 mark
      2
      Where does \(y=\tan x\) have asymptotes (0° to 360°)? Type both angles.
      Guide me

      What is it asking? two angles

      What do we know? where \(\cos x=0\)

      Plan division by zero

      Show working
      1. 90° and 270°

      Answer 90, 270

      PartAnswer / workingMarksGuidance
      90 and 270B1
      Total: 1 mark
      3
      Exact value of \(\sin60^\circ\times\cos30^\circ\)?
      Guide me

      What is it asking? a fraction

      What do we know? both are \(\tfrac{\sqrt3}2\)

      Plan multiply

      Show working
      1. \(\tfrac{\sqrt3}2\times\tfrac{\sqrt3}2=\tfrac34\)

      Answer 3/4

      PartAnswer / workingMarksGuidance
      \(\tfrac34\)B1
      Total: 1 mark

      Can I do this?

      Anything unticked: revise that section first. All ticked: take the quiz.

      Quiz

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