Trigonometric Equations

Solve Equations Involving a Single Trig Function

Checking if a Number is a Solution

Determine whether = is a solution of the equation 4

1sin . Is = a solution?

2 6

Finding All Solutions of A Trig Equation

Remember, trigonometric functions are periodic. Therefore, there an infinite number of solutions to the equation. To list all of the answers, we will have to determine a formula.

Finding All Solutions of A Trig Equation

Tan = 1 tan-1(tan tan-1 (1) = /4 To find all of the solutions, we need to

remember that the period of the tangent function is .

Therefore, the formula for all of the solutions is is an integer

4k k

Finding All Solutions of A Trig Equation

cos = 0 cos-1 (cos ) = cos-1 0 The period for cos is 2. Therefore, the

formula for all answers is 0 ± 2k (k is an integer)

Finding All Solutions of A Trig Equation

1 1

3cos

2

3cos (cos ) cos

2

5 7 5, : 2

6 6 67

26

so Answers k

k

Solving a Linear Trig Equation

Solve

1 1

11 cos 0 2

21

cos 12

1cos 1

21

cos cos cos cos2

5,

3 3

Subtract from both sides

Divide by

Take inverse on both sides

Solving a Trig Equation

Solve the equation on the interval 0 ≤ θ ≤ 2

2

2

4cos 1

1cos 4

41

cos2

2 4 5, , , cos

3 3 3 3

Divide both sides by

Take square root of both sides

Take inverse of both sides

Solving a Trig Equation

Solve the equation on the interval 0 ≤ θ ≤ 2

1

1sin(2 )

21 5

2 sin 2 22 6 6

5

12 12

Solving a Trig Equation

In order to get all answers from 0 to 2 , it is necessary

to add 2 to the original answers and solve for the

remaining answers.

52 = 2 2 2

6 613 13 17 17

2 26 12 6 12

Solving a Trig Equation

The number of answers to a trig equation on the interval 0 ≤ θ ≤ 2will be double the number in front of θ. In other words, if the angle is 2 θ the number of answers is 4. If the angle is 3 θ the number of answers is 6. If the angle is 4 θ the number of answers is 8, etc. unless the answer is a quadrantal angle.

Solving a Trig Equation

Keep adding 2 to the answers until you have the needed angles.

Solving a Trig Equation

Solve the equation on the interval 0 ≤ θ ≤ 2

1 1

sin 3 118

sin sin 3 sin 118

3 318 2 2 189 4

3 318 18 9

Solving a Trig Equation

4 43

9 274 22 22

3 2 39 9 27

22 40 403 2 3

9 9 27

Solving a Trig Equation

Solve the equation on the interval 0 ≤ θ ≤ 2

4sec 6 2

4sec 8 sec 2

1 12 cos

cos 22

3 3

Solving a Trig Equation with a Calculator

sin θ = 0.4 sin-1 (sin θ) = sin-1 0.4 θ = .411, - .411 = 2.73

sec θ = -4 1/cos θ = -4 cos θ = -¼ cos-1 (cos θ) = cos-1 (-¼) θ = 1.82 Need to find reference angle because this is

a quadrant II answer.

Solving a Trig Equation with a Calculator

To find reference angle given a Quad II angle – answer ( – 1.82 = 1.32)

Now add to this answer ( + 1.32) θ = 4.46

Snell’s Law of Refraction

Light, sound and other waves travel at different speeds, depending on the media (air, water, wood and so on) through which they pass. Suppose that light travels from a point A in one medium, where its speed is v1, to a point B in another medium, where its speed is v2. Angle θ1 is called the angle of incidence and the angle θ2 is the angle of refraction.

Snell’s Law of Refraction

Snell’s Law states that

1 1

2 2

sin

sin

v

v

Snell’s Law of Refraction

1

2

is also known as the index of refractionv

v

Some indices of refraction are given in the table on page 512

Snell’s Law of Refraction

The index of refraction of light in passing from a vacuum into water is 1.33. If the angle of incidence is 40o, determine the angle of refraction.

Snell’s Law of Refraction

1

1

2 2

2 2

2 2

sin 401.33 1.33

sin

sin 40sin 40 1.33sin sin

1.33

sin 40sin 28.9

1.33

o

oo

oo

vtherefore

v

Solving Trig Equations

Tutorial Sample Problems Video Explanations