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Trigonometric Graphs

Trigonometric Graphs

Solving trigonometric equations

Solving Trig Equations Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Solving a trig equation
  4. Negative values and no solutions
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus E6.4: trigonometric equations
  • E6.4Solve \(\sin x=k\), \(\cos x=k\) and \(\tan x=k\) for \(0^\circ\le x\le360^\circ\), giving every solution.
  • E6.4Rearrange first, e.g. \(2\cos x+1=0\).
  • E6.4Handle negative values and spot equations with no solutions.

Before you start: Graphs of sin, cos and tan and rearranging.

Formula bank

Second solutionLearn itsin: \(180^\circ-x\); cos: \(360^\circ-x\); tan: \(x+180^\circ\)x = the calculator's answer.Use it when every trig equation.
Into rangeLearn itadd 360° (sin, cos) or 180° (tan) to a negative angleUse it when the calculator gives a negative angle.
No solutionsLearn it\(\sin x\) or \(\cos x\) = a number above 1 or below −1the calculator says Math ERROR.

Solving a trig equation

The calculator gives one angle. The graph shows there is usually a second one between 0° and 360°, found with the symmetry rule.

General methodHow to solve any trig equation for 0° to 360°

  1. Rearrange to \(\sin x=k\), \(\cos x=k\) or \(\tan x=k\).
  2. For sin or cos: if k is above 1 or below −1, there are no solutions.
  3. Calculator: \(x=\sin^{-1}k\) (or \(\cos^{-1}k\), \(\tan^{-1}k\)).
  4. Second angle: sin → 180° − first; cos → 360° − first; tan → first + 180°.
  5. If an angle is negative, add 360° (or 180° for tan) to bring it into range. Give answers to 1 d.p.
cos x = −½90180270360-1.5-1.0-0.50.51.01.5xy120°240°
The line y = −½ cuts cos x twice.

\(2\cos x+1=0\Rightarrow\cos x=-\tfrac12\). Calculator: 120°. Second: 360° − 120° = 240°.

Example Solve \(4\cos x+1=0\) for \(0^\circ\le x\le360^\circ\).

Rearrange, calculator, then 360° − first.

  1. \(\cos x=-0.25\)
  2. \(x=\cos^{-1}(-0.25)=104.48\ldots\)
  3. second: \(360-104.48=255.52\)

Answer 104.5° or 255.5°

Try it yourself

1
Solve \(\sin x=0.3\) (1 d.p.).
Show answer

Answer 17.5, 162.5

2
Solve \(2\tan x=5\) (1 d.p.).
Show answer

Answer 68.2, 248.2

Negative values and no solutions

Example Solve \(3\sin x+1=0\).

\(\sin x=-\tfrac13\); the calculator gives a negative angle, so move it into range.

  1. \(\sin^{-1}(-\tfrac13)=-19.47^\circ\)
  2. \(-19.47+360=340.5^\circ\)
  3. second: \(180-(-19.47)=199.5^\circ\)

Answer 199.5° or 340.5°

sin x = −⅓90180270360-1.5-1.0-0.50.51.01.5xy199.5°340.5°
Both answers are where sin is negative: 180°-360°.

\(2\sin x=3\) gives \(\sin x=1.5\): impossible, so there are no solutions.

Try it yourself

1
Solve \(\tan x=-1\).
Show answer

Answer 135, 315

2
How many solutions has \(3\cos x=4\)?
Show answer

Answer 0

Ways they test it

Type 1 Rearrange and solve

"Solve \(2\cos x+1=0\) for \(0^\circ\le x\le360^\circ\)." (3 marks)

Spot it a trig function with numbers around it.

  1. cos x = −½.
  2. 120°.
  3. 360° − 120° = 240°.

Type 2 After a sketch

"(a) Sketch \(y=\sin x\). (b) Solve \(5\sin x-2=0\)." (2 + 3 marks)

Spot it a sketch in part (a).

  1. sin x = 0.4.
  2. 23.6° and 180° − 23.6°.

For example \(\sin x=0.4\)

    Answer 23.6°, 156.4°

    Type 3 Exact values

    "Solve \(2\sin x-\sqrt3=0\)." (non-calculator)

    Spot it a surd: an exact value.

    1. \(\sin x=\tfrac{\sqrt3}2\) → 60°.
    2. 180° − 60° = 120°.

    For example \(\sin x=\tfrac{\sqrt3}2\)

      Answer 60°, 120°

      Type 4 A condition

      "\(\cos x=\tfrac12\) and x is reflex. Find x." (1 mark)

      Spot it 'reflex', 'obtuse', 'acute'.

      1. Find both angles.
      2. Keep the one that fits.

      For example 60° and 300°

        Answer 300°

        Mistakes to avoid

        • ✗ Only the calculator answer✓ both solutionsThe graph crosses twice.
        • ✗ cos: 180° − x✓ cos: 360° − xThat is the sin rule.
        • ✗ Giving −19.5°✓ move it into 0°-360°Out of range.
        • ✗ sin x = 1.5 → 'I'm wrong'✓ no solutionsA valid answer.

        Summary

        • Rearrange to sin x, cos x or tan x = k.
        • Calculator first angle; second from the symmetry rule.
        • Negative angles: add 360° (tan: 180°).
        • |k| > 1 for sin or cos: no solutions.

        If you see …

        • "for 0° ≤ x ≤ 360°"expect two answers.
        • "reflex" / "obtuse"choose the answer in that range.

        Practice

        Practice: easy, medium, hard

        Easy

        1
        Solve \(2\cos x+1=0\).
        Show answer

        Answer 120, 240

        2
        Solve \(\sin x=0.4\) (1 d.p.).
        Show answer

        Answer 23.6, 156.4

        3
        \(\cos x=\tfrac12\) and x is reflex. Find x.
        Show answer

        Answer 300

        Medium

        1
        Solve \(4\cos x+1=0\) (1 d.p.).
        Guide me

        What is it asking? two angles

        What do we know? cos x = −0.25

        Plan 360° − first

        Show working
        1. \(\cos^{-1}(-0.25)=104.5\)
        2. 360 − 104.5 = 255.5

        Answer 104.5, 255.5

        2
        Solve \(3\sin x+1=0\) (1 d.p.).
        Guide me

        What is it asking? two angles

        What do we know? sin x = −1/3: calculator −19.47°

        Plan 180° + 19.47°, 360° − 19.47°

        Show working
        1. 199.5° and 340.5°

        Answer 199.5, 340.5

        3
        Solve \(\sin x=\tfrac23\) (1 d.p.).
        Guide me

        What is it asking? two angles

        What do we know? \(\sin^{-1}\tfrac23\)

        Plan 180° − first

        Show working
        1. 41.8° and 138.2°

        Answer 41.8, 138.2

        Hard

        1
        Solve \(5\sin x-2=0\) (1 d.p.).
        Guide me

        What is it asking? two angles

        What do we know? sin x = 0.4

        Plan 180° − first

        Show working
        1. \(\sin x=0.4\)
        2. 23.6° and 156.4°

        Answer 23.6, 156.4

        PartAnswer / workingMarksGuidance
        sin x = 0.4M1
        23.6B1
        156.4B1
        Total: 3 marks
        2
        Solve \(2\sin x-\sqrt3=0\).
        Guide me

        What is it asking? two angles

        What do we know? \(\sin x=\tfrac{\sqrt3}2\)

        Plan exact value 60°

        Show working
        1. 60° and 180° − 60° = 120°

        Answer 60, 120

        PartAnswer / workingMarksGuidance
        60B1
        120B1
        Total: 2 marks
        3
        Solve \(4\cos x+3=0\) (1 d.p.).
        Guide me

        What is it asking? two angles

        What do we know? cos x = −0.75

        Plan 360° − first

        Show working
        1. \(\cos^{-1}(-0.75)=138.6\)
        2. 360 − 138.6 = 221.4

        Answer 138.6, 221.4

        PartAnswer / workingMarksGuidance
        cos x = −0.75M1
        138.6B1
        221.4B1
        Total: 3 marks

        Can I do this?

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

        Quiz

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