Solving Trig Equations (Cambridge (CIE) IGCSE International Maths: Extended): Revision Note

Solving trig equations

What are trig equations?

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

• 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. $0°\le x\le 360°$

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 $\sin x=0.5$ for $0°\le x\le 360°$

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

• 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 $\sin x=0.5$ for $0°\le x\le 360°$

    • Sketch the graph $y=\sin x$ for $0°\le x\le 360°$

    • Draw on $\sin \left(30\right)=0.5$

    • By the symmetry, the new value of $x$ is $180°-30°=150°$

    • The solutions are $30°$ or $150°$

• Check the solutions

  • E.g. For the equation $\sin x=0.5$ for $0°\le x\le 360°$

    • Substitute $x=30°$ and $x=150°$ in to the calculator

    • $\sin \left(30\right)$ and $\sin \left(150\right)$ both give a value of $0.5$, so are correct

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

  • 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 $\cos x=0.5$ for $0°\le x\le 360°$

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

• 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 $\cos x=0.5$ for $0°\le x\le 360°$

    • Sketch the graph $y=\cos x$ for $0°\le x\le 360°$

    • By the symmetry, the new value of $x$ is $360°-60°=300°$

    • The solutions are $60°$ or $300°$

• Check the solutions

  • E.g. For the equation $\cos x=0.5$ for $0°\le x\le 360°$

    • Substitute $x=60°$ and $x=300°$ in to the calculator

    • $\cos \left(60\right)$ and $\cos \left(300\right)$ both give a value of $0.5$ so are correct

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

  • 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 $\tan x=1$ for $0°\le x\le 360°$

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

• 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 $\tan x=1$ for $0°\le x\le 360°$

    • Sketch the graph $y=\tan x$ for $0°\le x\le 360°$

    • By the periodic nature, the new value of $x$ is $45°+180°=225°$

• Check the solutions

  • E.g. For the equation $\tan x=0.5$ for $0°\le x\le 360°$

    • Substitute $x=45°$ and $x=225°$ in to the calculator

    • $\tan \left(45\right)$ and $\tan \left(225\right)$ both give a value of $1$ so are correct

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

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

  • 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