Transcript 5.4 Equations and Graphs of Trigonometric Functions

5.4 Equations and Graphs of Trigonometric Functions
Math 30-1
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Determine the angle given the ratio. 0  x  360
Determine the solution(s) for the trigonometric equation cos x 
3
.
2
We seek the angle (the value of x) for which the cosine gives the ratio
Graphically
y
Reference Angle
3.
2
3
2
The reference angle for cos x 
Quadrant I
Quadrant IV
3
2
x = 30°
x = 150°
Two solutions: 30° and 150°.
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Multiple Solutions
When the domain of the function is restricted (0° ≤ x < 360°), the solutions
to the trigonometric equation must occur within that given restriction.
When the domain of the function is not restricted, often there are
multiple solutions.
The solutions repeat
themselves in multiples of
360° from each original
solution.
The general solutions to
the equation cos x 
3
2
are x = 30° + 360°n or
x = 150° + 360°n, n ϵ I
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Determine the solutions for the trigonometric equation
3 tan2 x  1  0 for the interval 0  x  360
Method 1: Solve Graphically
2
Graph the related function y  3 tan x  1
The solutions are the xintercepts of the graph of
the related function.
(0, 150)
(0, 30)
(0, 210)
(0, 330)
Math 30-1
The solutions for the
interval 0° ≤ x < 360° are
x = 30°, 150°, 210°, 330°.
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Determine the solutions for the trigonometric equation
3 tan2 x  1  0 for the interval 0  x  360
Method 2: Solve algebraically
3 tan x  1  0
3 tan2 x  1
1
2
tan x 
3
tan x 
2
tan x  
30°
1
3
210°
Quadrant I
Quadrant III
x = 30°
x = 150°
1
3
tan x  
1
150°
3
330°
Quadrant I
Quadrant III
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x = 30°
x = 150°
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Solving Trigonometric Equations x , radians
2sin2 x + 3sin x + 1 = 0
(2sin x + 1)(sin x + 1) = 0
2sin x + 1 = 0
Reference
Angle
1
sin x 
2


sin x + 1 = 0
sin x  1
6
7 11
x=
,
6
6
7
x
 2n , n  I
6
x
3
2
x  3.67  2n , 4.71  2n ,
3
x
 2n , n  I
or 5.76  2n ,
2
nI
11
x
 2n , n  I
6
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Interpreting Graphs to Find Solutions
The diagram below shows the graphs of two trig functions
y = 4sin2x and y = 6sinx + 2, defined for 0 ≤ x ≤ 2.
Describe how you could use this graph to estimate the
solution to the equation (4sin2x) (6sinx + 2) = 0 for 0 ≤ x ≤ 2.
y = 6sinx + 2
y= 4sin2x
The solutions will be the points where the graphs intersect
the x-axis.
Therefore, the solutions are:
x = 0, 3.14, 6.28, 3.48, and 5.94
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Interpreting Graphs to Find Solutions
Describe how you could use this graph to estimate the
solution to the equation 4sin2x = 6sinx + 2 for 0 ≤ x ≤ 2.
y = 6sinx + 2
y= 4sin2x
The solutions will be the points where the graphs intersect.
Therefore, the solutions are:
x = 3.426 and 5.999.
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Interpreting Graphs to Find Solutions
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Use a graph to solve the equation sin x  x.
3
sin x 
1
x0
3
Therefore, the solutions are:
x = 0 and 2.2789
Math 30-1
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Using Technology to Find Solutions
Determine the lowest possible value of x, to the
x
nearest tenth for which  2sin 3x  0.
2
The smallest x-value is -2.9.
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