Trigonometric Equations (SQA National 5 Maths): Revision Note

Solving trigonometric equations

What are trigonometric equations?

• Trigonometric 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 periods and related angles to find the others

  • The solutions you find 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 if you know them)

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

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

• Use the symmetry of the sine function and related angles to find other solutions

  • If $x=30°$ is a solution, then by symmetry $x=180-30=150°$ is another solution

  • If necessary you can also use the period of $\sin x$ to find additional solutions

    • Adding or subtracting 360° to a solution gives another solution

• You can use a calculator to 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

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 if you know them)

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

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

• Use the symmetry of the cosine function and related angles to find other solutions

  • If $x=60°$ is a solution, then by symmetry $x=360-60=300°$ is another solution

  • If necessary you can also use the period of $\cos x$ to find additional solutions

    • Adding or subtracting 360° to a solution gives another solution

• You can use a calculator to 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

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 if you know them)

  • E.g. For the first solution of the equation $\tan x=1$ for $0°\le x\le 360°$

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

• Use the period of the tan function to find other solutions

  • Adding or subtracting 180° to a solution gives another solution

  • If $x=45°$ is a solution, then by symmetry $x=45+180=225°$ is another solution

• You can use a calculator to 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

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 $\sin x=0.5$

  • 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

  • E.g $x={\sin }^{-1}\left(-0.5\right)=-30°$

• In that case, use the period to find a positive solution

  • The period of $\sin x$ is 360°

  • So $-30+360=330°$ is another solution

• Once you have a positive solution, you can use related angles to find any other positive solutions in the interval