Transcript Solving a Trig Equation

Trigonometric Equations
Solve Equations
Involving a Single Trig
Function
Checking if a Number is a Solution
Determine whether  =

4
is a solution of the equation
1

sin   . Is  = a solution?
2
6
Finding All Solutions of A Trig
Equation
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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
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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

4
k
k is an integer
Finding All Solutions of A Trig
Equation
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cos  = 0
cos-1 (cos ) = cos-1 0
0
The period for cos is 2. Therefore, the
formula for all answers is 0 ± 2k (k is an
integer).
Finding All Solutions of A Trig
Equation
3
cos   
2

3
cos (cos  )  cos  

2


5 7
5

,
so
Answers :
 2 k
6 6
6
7
 2 k
6
1
1
Solving a Linear Trig Equation
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Solve 1  cos   1
2
1
 cos   
2
1
cos  
2
0    2
Subtract  1 from both sides
Divide by  1
1
cos  cos  )  cos
Take inverse cos on both sides
2
 5
 ,
3 3
1
1
Solving a Trig Equation
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Solve the equation on the interval 0 ≤ θ ≤ 2
4 cos 2   1
1
2
cos  
4
1
cos   
2
 2 4 5
 , , ,
3 3 3 3
Divide both sides by 4
Take square root of both sides
Take inverse cos of both sides
Solving a Trig Equation
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Solve the equation on the interval 0 ≤ θ ≤ 2
1
sin(2 ) 
2
1 1
2  sin
2
2 


6

12
5
2 
6
5

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.
2 =

 2
6
13
2 
6
13

12
5
2 
 2
6
17
17
2 

6
12
Solving a Trig Equation
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The number of answers to a trig equation on
the interval 0 ≤ θ ≤ 2 will 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
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Keep adding 2 to the answers until you
have the needed angles.
Solving a Trig Equation
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Solve the equation on the interval 0 ≤ θ ≤ 2
 

sin  3    1
18 

 

1 
sin  sin  3     sin 1 1
18  
 
3 



18 2
9 
3 

18 18
3 



2 18
4
3 
9
Solving a Trig Equation
4
3 
9
4
3 
 2
9
22
3 
 2
9
4

27
22
3 
9
40
3 
9
22

27
40

27
Solving a Trig Equation
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Solve the equation on the interval 0 ≤ θ ≤ 2
4sec   6  2
4sec   8
1
 2
cos 
 

3

sec   2
1
cos   
2
2

3
Solving a Trig Equation with a
Calculator
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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
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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
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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
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Snell’s Law states that
sin 1 v1

sin  2 v2
Snell’s Law of Refraction
v1
is also known as the index of refraction
v2
Some indices of refraction are given in the table
on page 512
Snell’s Law of Refraction
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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
v1
 1.33
v2
o
therefore
sin 40
 1.33
sin  2
sin 40  1.33sin  2
sin 40o
 sin  2
1.33
 sin 40o 
sin 
  2
 1.33 
28.9   2
o
1
o
Solving Trig Equations
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