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Chapter 66.5 Trigonometric equations

Dr .Hayk MelikyanDepartmen of Mathematics and CS

melikyan@nccu.edu

• Find all solutions of a trigonometric equation.• Solve equations with multiple angles.• Solve trigonometric equations quadratic in

form.• Use factoring to separate different functions in

trigonometric equations.• Use identities to solve trigonometric equations.• Use a calculator to solve trigonometric

equations

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Trigonometric Equations and Their Solutions

A trigonometric equation is an equation that contains a trigonometric expression with a variable, such as sin x.

The values that satisfy such an equation are its solutions. (There are trigonometric equations that have no solution.)

When an equation includes multiple angles, the period of the function plays an important role in ensuring that we do not leave out any solutions.

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• To solve an equation containing a single trigonometric function:

• Isolate the function on one side of the equation.

sinx = a (-1 ≤ a ≤ 1 )

cosx = a (-1 ≤ a ≤ 1 )

tan x = a ( for any real a )

• Solve for the variable.

Equations Involving a Single Trigonometric Function

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y = cos x

x

y

1

–1

y = 0.5

–4 2–2 4

cos x = 0.5 has infinitely many solutions for – < x <

y = cos x

x

y

1

–1

0.5

2

cos x = 0.5 has two solutions for 0 < x < 2

Trigonometric Equations

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Solve the equation: 3 sin x 2 5 sin x 1.

Solution The equation contains a single trigonometric function, sin x.

Step 1 Isolate the function on one side of the equation. We can solve for sin x by collecting all terms with sin x on the left side, and all the constant terms on the right side.

3 sin x 2 5 sin x 1 This is the given equation.

3 sin x 5 sin x 2 5 sin x 5 sin x – 1 Subtract 5 sin x from both sides.

sin x -1/2

Divide both sides by 2 and solve for sin x.

2 sin x 1 Add 2 to both sides.

2 sin x 2 1 Simplify.

Text Example

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Solve the equation: 2 cos2 x cos x 1 0, 0 x 2.

The solutions in the interval [0, 2) are /3, , and 5/3.

Solution The given equation is in quadratic form 2t2 t 1 0 with t cos x. Let us attempt to solve the equation using factoring.

2 cos2 x cos x 1 0 This is the given equation.

(2 cos x 1)(cos x 1) 0 Factor. Notice that 2t2 + t – 1 factors as (t – 1)(2t + 1).

cos x 1/2

2 cos x 1 cos x 1 Solve for cos x.

2 cos x 1 0 or cos x 1 0

Set each factor equal to 0.

Text Example

x x 2 x

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Example

cos29cos7

Solve the following equation:

Solution:

n2

5,3,

1cos

9cos9

cos29cos7

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Example

Solve the equation on the interval [0,2)

Solution:

3

3

2tan

3

7

3

6

7

62

3

3

2tan

and

and

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Example

Solve the equation on the interval [0,2)

Solution:03cos2cos2 xx

0

0

1cos3cos

01cos03cos

0)1)(cos3(cos

03cos2cos2

x

xsolutionno

xx

xx

xx

xx

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Example

Solve the equation on the interval [0,2)

Solution:

3

5,

3

2

1cos

1cos2

sincossin2

sin2sin

x

x

x

xxx

xx

xx sin2sin

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Example: Finding all Solutions of a Trigonometric Equation Solve the equation: Step 1 Isolate the function on one side of the

equation.

5sin 3sin 3. x x

5sin 3sin 3x x

5sin 3sin 3sin 3sin 3 x x x x

2sin 3x

3sin

2x

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Example: Finding all Solutions of a Trigonometric Equation (continued) Solve the equation: Step 2 Solve for the variable.

5sin 3sin 3. x x

3sin

2x

Solutions for this equation in are: 0,2

The solutions for this equation are:

2,

3 3

22 , 2

3 3n n

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Solving an Equation with a Multiple Angle

Solve the equation: tan 2 3,0 2 . x x

tan 33

Because the period is all solutions for this equation are given by

,

23

x n

6 2n

x

0n 06 2 6

x

1n3 4 2

6 2 6 6 6 3x

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Solving an Equation with a Multiple Angle

Solve the equation: tan 2 3,0 2 . x x

Because the period is all solutions for this equation are

given by

,.

6 2 nx

2n

3n

2 6 76 2 6 6 6

x

3 9 10 56 2 6 6 6 3

x

In the interval , the solutions are:2 7 5

, , , and .6 3 6 3 0,2

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Solving a Trigonometric Equation Quadratic in Form Solve the equation: 24cos 3 0, 0 2 . x x

24cos 3 0x 24cos 3x

2 3cos

4x

3 3cos

4 2x

The solutions in the interval for this equation are:

5 7 11, , , and .

6 6 6 6

0,2

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Using Factoring to Separate Different Functions

Solve the equation:sin tan sin , 0 2 . x x x xsin tan sinx x xsin tan sin 0x x x

sin (tan 1) 0x x

sin 0x

0 x x

tan 1 0x

tan 1x

5

4 4 x x

The solutions for this equation in the interval are:5

0, , , and .4 4

0,2

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Using an Identity to Solve a Trigonometric Equation

Solve the equation:cos2 sin 0, 0 2 . x x xcos2 sin 0x x

21 2sin sin 0x x 22sin sin 1 0x x

(2sin 1)(sin 1) 0x x 2sin 1 0x

2sin 1x 1

sin2

x

7 11

6 6 x x

sin 1 0x

sin 1x

x

The solutions in the

interval are7 11

, , and .6 6

0,2

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Solving Trigonometric Equations with a Calculator

Solve the equation, correct to four decimal places, for0 2 .x

tan 3.1044x 1tan (3.1044)x

1.2592x

tanx is positive in quadrants I and III

In quadrant I

In quadrant III

1.2592x

1.2592x

4.4008

The solutions for this equation are 1.2592 and 4.4008.

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Using a Calculator to Solve Trigonometric Equations Solve the equation, correct to four decimal

places, for0 2 .x

sin 0.2315x 1sin ( 0.2315)x

0.2336x

Sin x is negative in quadrants III and IV

In quadrant III

In quadrant IV 2 1.2592

6.0496

x

x

The solutions for this equation are 3.3752 and 6.0496.

0.2336

3.3752

x

x