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Lesson 5.3

Solving Trigonometric Equations

Solving Trigonometric Equations

To solve trigonometric equations:

Use standard algebraic techniques learned in Algebra II.

Look for factoring and collecting like terms.

Isolate the trig function in the equation.

Use the inverse trig functions to assist in determining

solutions.

Solving Trigonometric Equations

For all problems,

The solution interval

Will be

[0, 2)

You are responsible for checking your solutions back into the original problem!

Solving Trigonometric Equations

Solve:

Step 1: Isosolate cos x using algebraic skills.

Step 2: Determine in which quadrants cosine is positive. Use the inverse

function to assist by finding the angle in Quad I first. Then use that angle

as the reference angle for the other quadrant(s).

Note: cosine is positive in

Quad I and Quad IV.

QI

QIV

Note: The reference angle is /3.

Note: Since there is a , all four quadrants

hold a solution with /4 being the reference

angle.

Solving Trigonometric EquationsSolve:

Step 1:

Step 2:

QIV

QIII

Q1

QII

Note: There is no solution here because 2

lies outside the range for cosine.

Solving Trigonometric EquationsSolve:

Step 1:

Step 2:

Solving Trigonometric Equations

Solve:

Factor the quadratic equation.

Set each factor equal to zero.

Solve for sin x

Determine the correct quadrants

for the solution(s).

Solving Trigonometric Equations

Solve:

Replace sin2x with 1-cos2x

Distribute

Combine like terms.

Multiply through by – 1.

Factor.

Set each factor equal to zero.

Solve for cos x.

Determine the solution(s).

Solving Trigonometric Equations

Solve:

Square both sides of the equation

in order to change sine into terms

of cosine giving only one trig

function to work with.

FOIL or Double Distribute

Replace sin2x with 1 – cos2x

Set equation equal to zero since it is a

quadratic equation.

Factor

Set each factor equal to zero.

Solve for cos x

X

Determine the solution(s).

It is removed because it does not

check in the original equation.

Why is 3/2 removed as a solution?

Solving Trigonometric Equations

Solve:

No algebraic work needs to be done because cosine is already by itself.

Remember, 3x refers to an angle and one cannot divide by 3 because it

is cos 3x which equals ½.

Solution:

Since 3x refers to an angle, find the angles whose cosine value is ½.

Now divide by 3 because it is angle equaling angle.

Notice the solutions do not exceed 2. Therefore,

more solutions may exist.

Return to the step where you have 3x equaling

the two angles and find coterminal angles for

those two.

Divide those two new angles by 3.

Solving Trigonometric Equations

The solutions still do not exceed 2.

Return to 3x and find two more

coterminal angles.

Divide those two new angles by 3.

The solutions still do not exceed 2.

Return to 3x and find two more

coterminal angles.

Divide those two new angles by 3.

Notice that 19/9 now exceeds 2 and

is not part of the solution.

Therefore the solution to cos 3x = ½ is

Solving Trigonometric Equations

What you should know:

- How to use algebraic techniques to solve
- trigonometric equations.

- How to solve quadratic trigonometric equations
- by factoring or the quadratic formula.

- How to solve trigonometric equations involving
- multiple angles.

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