Solving Trigonometric Equations

MATH 109 - PrecalculusS. Rook

Overview

• Section 5.3 in the textbook:– Basics of solving trigonometric equations– Solving linear trigonometric equations– Solving quadratic trigonometric equations– Solving trigonometric equations with multiple

angles– Solving other types of trigonometric equations– Approximate solutions to trigonometric equations

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Basics of Solving Trigonometric Equations

Basics of Solving Trigonometric Equations

• To solve a trigonometric equation when the trigonometric function has been isolated:– e.g.– Look for solutions in the interval 0 ≤ θ < period using the unit

circle• Recall the period is 2π for sine, cosine, secant, & cosecant and

π for tangent & cotangent• We have seen how to do this when we discussed the circular

trigonometric functions in section 4.2– If looking for ALL solutions, add period n to each individual ∙

solution• Recall the concept of coterminal angles

4

2

3sin

Basics of Solving Trigonometric Equations (Continued)

– We can use a graphing calculator to help check (NOT solve for) the solutions• E.g. For , enter Y1 = sin x, Y2 = , and look

for the intersection using 2nd → Calc → Intersect

5

2

3sin

2

3

Basics of Solving Trigonometric Equations (Example)

Ex 1: Find all solutions and then check using a graphing calculator:

6

3tan

Solving Linear Trigonometric Equations

Solving Linear Equations

• Recall how to solve linear algebraic equations:– Apply the Addition Property of Equality• Isolate the variable on one side of the equation• Add to both sides the opposites of terms not associated

with the variable

– Apply the Multiplication Property of Equality• Divide both sides by the constant multiplying the

variable (multiply by the reciprocal)

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Solving Linear Trigonometric Equations

• An example of a linear equation:• Solving trigonometric linear (first

degree) equations is very similar EXCEPT we:– Isolate a trigonometric function of an angle instead of a

variable• Can view the trigonometric function as a variable by making a

substitution such as • Revert to the trigonometric function after isolating the

variable

– Use the Unit Circle and/or reference angles to solve

9

4

82

352

3553

x

x

x

xx

sinx

Solving Linear Trigonometric Equations (Example)

Ex 2: Find all solutions:

10

xx cos1cos

Solving Quadratic Trigonometric Equations

Solving Quadratic Trigonometric Equations

• Recall a Quadratic Equation (second degree) has the format– One side MUST be set to zero

• Common methods used to solve a quadratic equation:– Factoring• Remember that the process of factoring converts a sum

of terms into a product of terms– Usually into two binomials

– Quadratic Formula12

02 cbxax

Solving Quadratic Trigonometric Equations (Continued)

• The same methods can be used to solve a quadratic trigonometric equation:– Substituting a variable for a trigonometric

function is acceptable so long as there is only one trigonometric function present in the equation• e.g. Let y = tan x

– Be aware of extraneous solutions if fractions are present• Those solutions which cause the denominator to equal

0

13

0101tan 22 yx

Solving Quadratic Trigonometric Equations (Example)

Ex 3: Solve in the interval 0 ≤ x < 2π:

a)

b)

c)

14

01coscos2 2 xx

0tansintan xxx

04sec3 2 x

Trigonometric Equations with Two Different Trigonometric Functions• Be aware when a quadratic trigonometric

equation exists with two DIFFERENT trigonometric functions– Not like Example 3c because after factoring out

tan x, the equation became two linear trigonometric equations

– Recall how we handled two different trigonometric functions in section 5.1

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Trigonometric Equations with Two Different Trigonometric Functions (Continued)

• If we have two different trigonometric functions raised to the first power:– Square both sides and apply Pythagorean

identities to simplify the equation• E.g.

– Recall that when we square both sides of an equation some of the potential solutions will not check into the original equation• MUST check all solutions into the original problem• Discard those solutions that do not check

16

xxxxxxxx cossin21coscossin2sin1cossin 22

Trigonometric Equations with Two Different Trigonometric Functions (Example)

Ex 4: Solve in the interval 0 ≤ x < 2π:

a)

b)

c)

17

05cos4sin4 2 xx

1cossin xx

1cotcsc xx

Solving Trigonometric Equations with Multiple Angles

Solving Trigonometric Equations with Multiple Angles

• A trigonometric equation with a multiple angle has the form kx where k ≠ 1 (a single-angle trigonometric function otherwise)

• To solve a trigonometric equation with multiple-angles e.g. 1 + cos 3x = 3⁄2: – Isolate the trigonometric function either by

solving for it or applying a quadratic strategy:• e.g. cos 3x = ½

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Solving Trigonometric Equations with Multiple Angles (Continued)– Find all solutions in the interval of [0, period)• e.g.

– Isolate the variable:• e.g.

– If necessary, let n vary to find all solutions in the interval [0, 2π):• e.g.

20

nxnx 2

3

53,2

33

3

2

9

5,

3

2

9

nx

nx

9

17,

9

13,

9

11,

9

7,

9

5,9

x

Solving Trigonometric Equations with Multiple Angles (Example)

Ex 5: Find all solutions in the interval [0, 2π):

21

2

24sin x

Other Types of Trigonometric Equations

Trigonometric Equations and the Sum & Difference Formulas

• Recall the sum and difference formulas– Valid in both directions

• Given a trigonometric equation involving the right-hand side of a sum or difference formula:– Condense into the left-hand side of the formula• e.g.

– Use previously discussed strategies to solve

23

15sin123sin12sin3cos2cos3sin xxxxxxx

Trigonometric Equations and Multiple-Angle Formulas

• Recall the double-angle and half-angle formulas– We can use either the left or right sides of these

formulas

• Overall goal is to isolate the trigonometric function

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Other Types of Trigonometric Equations (Example)

Ex 6: Solve in the interval [0, 2π):

a)

b) sin 6x + sin 2x = 0

c) 4 sin x cos x = 1

d)25

13

sin3

sin

xx

0sin2

cos xx

Approximate Solutions to Trigonometric Equations

Approximate Solutions to Trigonometric Equations

• More often than not we run into solutions of trigonometric equations that are NOT one of the special values on the unit circle

• Solve as normal until the trigonometric function is isolated

• Calculate the reference angle• Use the reference angle AND the sign of the

value of the trigonometric function to estimate the solutions in the interval [0, period)

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Approximate Solutions to Trigonometric Equations (Example)Ex 7: Find all solutions in the interval [0, 2π) –

use a calculator to estimate:

a)

b)

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4tan21tan8

013cos83cos4 2

Summary

• After studying these slides, you should be able to:– Solve linear trigonometric equations– Solve quadratic trigonometric equations– Solve trigonometric equations with multiple angles– Solve other types of trigonometric equations including sum &

difference formulas, double-angle & half-angle formulas – Approximate the solutions to trigonometric equations

• Additional Practice– See the list of suggested problems for 5.3

• Next lesson– Law of Sines (Section 6.1)

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