Trigonometric Equations Solve Equations Involving a Single Trig Function.

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Transcript of Trigonometric Equations Solve Equations Involving a Single Trig Function.
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 tan1(tan tan1 (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 cos1 (cos ) = cos1 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 sin1 (sin θ) = sin1 0.4 θ = .411,  .411 = 2.73
sec θ = 4 1/cos θ = 4 cos θ = ¼ cos1 (cos θ) = cos1 (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