3.12.2 Solving trigonometric equations

3.12.2 Solving trigonometric equations

Solving trigonometric equations involves finding the values of the variables that satisfy the given trigonometric equation.

Here are some general steps you can follow:

• Identify the trigonometric function(s) involved in the equation.

• Use trigonometric identities to simplify the equation, if possible.

• Try to isolate the trigonometric function on one side of the equation.

• Use inverse trigonometric functions to solve for the variable.

• Check your solution(s) by substituting them back into the original equation.

Worked example:

Solve for \( x \) : \( sinx=\frac{1}{2} \)

Identify the trigonometric function involved

\( sin(x) \)

Isolate the trigonometric function

\( sin(x)=\frac{1}{2} \)

\( x = sin^{-1}(\frac{1}{2}) \)

Use inverse trigonometric functions to solve for \( x \)

\( x = \frac{ \pi }{6} \) or \( x = \frac{5\pi }{6} \)

Check your solutions

\( sin(\frac{\pi }{6}) = \frac{1}{2} \) and \( sin \frac{5\pi }{6} = \frac{1}{2} \) (so both solutions are correct.)

Worked example:

Solve for \( x \) : \( 2sin(x) + cos(x) = 1 \)

Identify the trigonometric functions involved

\( sin(x) \) and \( cos(x) \)

Isolate the trigonometric functions

\( 2sin(x) + cos(x) = 1 \)

\( sin(x) = \frac{1-cos(x)}{2} \)

Use inverse trigonometric functions to solve for \( x \)

Let \( u = cos(x) \), then \( sin(x) = \frac{1-u}{2} \)

Substituting into the original equation gives

\( 2(\frac{1-u}{2} ) + u = 1 \)

which simplifies to

\( u = \frac{1}{3} \)

Therefore,

\( cos(x) = \frac{1}{3} \)

\( x = cos^{-1}\frac{1}{3} \)

Check your solution

\( 2sin(cos^{-1}(\frac{1}{3})) + cos(cos^{-1}(\frac{1}{3}))=1 \)

(so the solution is correct)

Test yourself

Question 1:

On the diagram, sketch the graph of \( y=tan x \) for \( 0°≤x≤360° \)

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Question 2:

Solve the equationtan \( tan x =2 \) for \( 0°≤x≤360° \)

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Question 3:

Solve \( 3tan x =-4 \) for \( 0°≤x≤360° \)

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Question 4:

On the diagram, sketch the graph of \( y=cos x \) for \( 0°≤x≤360° \).

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Solve the equation \( 4 cos x +2=3 \) for \( 0°≤x≤360° \)

[3]

Question 5:

Solve the equation \( 5tan x =-7 \) for \( 0°≤x≤360° \)

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