Solving Trigonometric Equations

TrigonometryMATH 103

S. Rook

Overview

• Section 6.1 in the textbook:– Solving linear trigonometric equations– Solving quadratic 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

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4

82

352

3553

x

x

x

xx

sinx

Solving Linear Trigonometric Equations (Example)

Ex 2: Find i) θ, 0° ≤ θ < 360° ii) all degree solutions

10

03cos2

Solving Linear Trigonometric Equations (Example)

Ex 3: Find i) t, 0 ≤ t < 2π ii) all radian solutions

11

tt sin5sin33

Solving Linear Trigonometric Equations (Example)

Ex 4: Find i) θ, 0° ≤ θ < 360° ii) all degree solutions – use a calculator to estimate:

a)

b)

c)12

4cos21cos8

sin24sin

12sin33

Solving Quadratic Trigonometric Equations

Solving Quadratic 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 Formula14

02 cbxax

Factoring a Quadratic

• To attempt factoring :– Always look for a GCF (greatest common factor)

• If present, factoring out the GCF simplifies the problem– Find two numbers that multiply to a·c AND add to b

• Only using the coefficients (numbers)– If a = 1, we have an easy trinomial

• Can immediately write as two binomials– If a ≠ 1, we have a hard trinomial

• Expand the trinomial into four terms• Use grouping• Alternatively, can also use “Guess and Check”

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02 cbxax

Solving Quadratic Equations Using the Quadratic Formula

• An equation in the format can also be solved using the Quadratic Formula:

• To solve a quadratic equation using the Quadratic Formula:– Set one side of the quadratic equation to zero– Plug the values of a, b, and c into the Quadratic Formula

• a, b, and c are all NUMBERS– Simplify

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02 cbxax

a

acbbx

2

42

Solving Quadratic Trigonometric Equations

• Solving quadratic trigonometric equations is very similar EXCEPT we:– Attempt to factor or use the Quadratic Formula on a

trigonometric function 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– Be aware of extraneous solutions if fractions OR functions

other than sine or cosine enter into the equation

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cosx

Solving Quadratic Trigonometric Equations (Example)

Ex 5: Find i) x, 0 ≤ x < 2π ii) all radian solutions

a)

b)

c)

18

0cotcot2 xx

3sin7sin2 2

0tansintan xxx

Solving Quadratic Trigonometric Equations (Example)

Ex 6: Find i) θ, 0° ≤ θ < 360° ii) all degree solutions – use a calculator to estimate:

19

2sin1sin

Additional Examples

Ex 7: In a) find all exact degree solutions and in b) find all exact radian solutions

a)

b)

20

2

350sin A

2

1

12cos

A

Summary

• After studying these slides, you should be able to:– Solve Linear Trigonometric Equations– Solve Quadratic Trigonometric Equations

• Additional Practice– See the list of suggested problems for 6.1

• Next lesson– More on Trigonometric Equations (Section 6.2)

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