Solving Trig Equations (Cambridge International IGCSE Maths)

Solving trig equations

What are trig equations?

• Trig equations are equations involving $\mathrm{sin}x$, $\mathrm{cos}x$ and $\mathrm{tan}x$

• They often have multiple solutions

  • A calculator gives the first solution
  • You need to use trig graphs to find the others
  • The solutions must lie in the interval (range) of $x$ given in the question, e.g. $0^{\circ}\leq x\leq360^{\circ}$

How do I solve sin x = ...?

• Find the first solution of the equation by taking the inverse sin function on your calculator (or using an exact trig value)

  • E.g. For the first solution of the equation $\mathrm{sin}x=0.5$ for $0^{\circ}\leq x\leq360^{\circ}$

    • This gives $x = \sin ^{- 1} ( 0.5 ) = 30^{\circ}$

• Then sketch the sine graph for the given interval

  • Identify the first solution on the graph

  • Use the symmetry of the graph to find additional solutions

  • E.g. For the equation $\mathrm{sin}x=0.5$ for $0^{\circ}\leq x\leq360^{\circ}$

    • Sketch the graph $y=\mathrm{sin}x$ for $0^{\circ}\leq x\leq360^{\circ}$
    • Draw on $\sin ( 30 ) = 0.5$
    • By the symmetry, the new value of $x$ is $180^{\circ}-30^{\circ}=150^{\circ}$
    • The solutions are $30^{\circ}$ or $150^{\circ}$

• Check the solutions

  • E.g. For the equation $\mathrm{sin}x=0.5$ for $0^{\circ}\leq x\leq360^{\circ}$

    • Substitute $x=30^{\circ}$ and $x=150^{\circ}$ in to the calculator
    • $\sin ( 30 )$ and $\sin ( 150 )$ both give a value of $0.5$, so are correct

• In general, if $x$ is an acute solution to $\mathrm{sin}x=...$

  • Then $180-x$ is an obtuse solution to the same equation

How do I solve cos x = ...?

• Find the first solution of the equation by taking the inverse cos function (or using an exact trig value)

  • E.g. For the first solution of the equation $\mathrm{cos}x=0.5$ for $0^{\circ}\leq x\leq360^{\circ}$

    • This gives $x = \cos ^{- 1} ( 0.5 ) = 60^{\circ}$

• Then sketch the cosine graph for the given interval

  • Identify the first solution on the graph

  • Use the symmetry of the graph to find additional solutions

  • E.g. For the equation $\mathrm{cos}x=0.5$ for $0^{\circ}\leq x\leq360^{\circ}$

    • Sketch the graph $y=\mathrm{cos}x$ for $0^{\circ}\leq x\leq360^{\circ}$
    • By the symmetry, the new value of $x$ is $360^{\circ}-60^{\circ}=300^{\circ}$
    • The solutions are $60^{\circ}$ or $300^{\circ}$

• Check the solutions

  • E.g. For the equation $\mathrm{cos}x=0.5$ for $0^{\circ}\leq x\leq360^{\circ}$

    • Substitute $x=60^{\circ}$ and $x=300^{\circ}$ in to the calculator
    • $\cos ( 60 )$ and $\cos ( 300 )$ both give a value of $0.5$ so are correct

• In general, if $x$ is a solution to $\mathrm{cos}x=...$

  • Then $360-x$ is another solution to the same equation

How do I solve tan x = ...?

• Find the first solution of the equation by taking the inverse tan function (or using an exact trig value)

  • E.g. For the first solution of the equation $\mathrm{tan}x=1$ for $0^{\circ}\leq x\leq360^{\circ}$

    • This gives $x = \tan ^{- 1} ( 1 ) = 45^{\circ}$

• Then sketch the tangent graph for the given interval

  • Identify the first solution on the graph

  • Use the periodic nature of the graph to find additional solutions

  • E.g. For the equation $\mathrm{tan}x=1$ for $0^{\circ}\leq x\leq360^{\circ}$

    • Sketch the graph $y=\mathrm{tan}x$ for $0^{\circ}\leq x\leq360^{\circ}$
    • By the periodic nature, the new value of $x$ is $45^{\circ}+180^{\circ}=225^{\circ}$

• Check the solutions

  • E.g. For the equation $\mathrm{tan}x=0.5$ for $0^{\circ}\leq x\leq360^{\circ}$

    • Substitute $x=45^{\circ}$ and $x=225^{\circ}$ in to the calculator
    • $\tan ( 45 )$ and $\tan ( 225 )$ both give a value of $1$ so are correct

• In general, if $x$ is a solution to $\mathrm{tan}x=...$

  • Then $x+180$ is another solution to the same equation

How do I rearrange trig equations?

• Trig equations may be given in a different form

  • Equations may require rearranging first

    • E.g. $2\mathrm{sin}x-1=0$ can be rearranged to $\mathrm{sin}x=\frac{1}{2}$

  • They can then be solved as usual

What do I do if the first solution from my calculator is negative?

• Sometimes the first solution given by the calculator for $x$will be negative

  • Continue sketching the graph to the left of the $x$-axis to help
  • Then find solutions that lie in the interval given in the question

Exam Hint Make sure you know how to use the inverse functions (e.g. $\mathrm{sin}^{-1}$) on your calculator.

Worked Example Use the graph of $y=\mathrm{sin}x$ to solve the equation $\mathrm{sin}x=0.25$ for $0^{\circ}\leq x\leq360^{\circ}$.

Give your answers correct to 1 decimal place.

Answer:

Use a calculator to find the first solution Take the inverse sin of both sides

$x = \sin ^{- 1} ( 0.25 ) = 14.47751...$

Sketch the graph of $y=\mathrm{sin}x$

Mark on (roughly) where $x=14.48$ and $y=0.25$ would be

Draw a vertical line up to the curve Draw another line horizontally across to the next point on the curve Bring a line vertically back down to the x-axis

Find this value using the symmetry of the curve Subtract 14.48 from 180

$180-14.48=165.52$

Give both answers correct to 1 decimal place

Final answer: $x=14.5^{\circ}$or$x=165.5^{\circ}$