hprec8 3
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8.3: Algebraic Solutions of Trigonometric Equations
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Solve Trigonometric equations using Algebraic methods.
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Try ThisUsing your knowledge of Algebra and factoring, solve the equation:
2 0x x
0, 1x x
Check your answer.
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ExampleSolve the equation and check the solution:
2sin sin 0x x
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Important Idea
Always check your solution(s) in the original equation.
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Try ThisUsing your knowledge of Algebra, solve the equation:
2 3 9x
3x
Check your answer.
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ExampleSolve the equation and check the solution:
2cos 1 3
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Important IdeaSimilar methods are used to solve algebraic and trigonometric equations.
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ExampleSolve for all values:
…there are an infinite number of solutions.
2cos 3x
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DefinitionPrincipal Values in Degrees:
& tan x: 90 90x
cos x: 0 180x
Principal values insure unique solutions
sin x
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DefinitionPrincipal Values in Radians:
sin x & tan x: 2 2x
cos x: 0 x
Principal values insure unique solutions
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Try ThisSolve for the principal value:
2cos 1 0x
60x or3
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Try ThisSolve for the principal value in degrees:
30x
What is the answer in rads? 6
x
2sin 1 0x
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Try ThisSolve for the principal value in radians:
3sin 1x
.340x
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Important Idea
When you cannot get exact values for solutions, you must use the inverse trig functions on your calculator.
1sin x
1cos x
1tan x
•
•
•
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Try ThisSolve for the principal value in degrees:
8cos 5 0
51.318x
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ExampleSolve for all values in degrees:
8cos 5 0
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Try This
Solve: 2sin 5 0x
???????
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Important Idea
The value of sin x and cos x can never be greater than 1 or less than -1
Range:sin x : [-1,1]
cos x : [-1,1]
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Try ThisUsing your knowledge of Algebra and factoring, solve the equation:
2 2 0x x
1, 2x x
Check your answer.
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ExampleSolve the equation and check the solution:
2sin sin 2 0x x
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Try ThisSolve the equation and check the solution:
2cos cos 2 0x x 0x
Are there other solutions?
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Example
Find all solutions of 23sin sin 2 0x x
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Try This Solve for the principal values in radians:
22sin sin 1 0x x
6x
or
2
Why is not an answer?
3
2
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ExampleFind all solutions in radians:
2tan cos tanx x x
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Try ThisFind the principal value solutions in degrees:
2cos sin cosx x x
90x 90x or
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ExampleSolve:
210cos 3sin 9 0x x
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ExampleSolve:
2sec 5tan 2x x
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Important Idea
2 2sin cos 1x x 2 2sin 1 cosx x 2 2cos 1 sinx x
Trigonometric Identities:
2 2tan 1 secx x 2 21 cot cscx x
See section 6.5 of your text for additional trig identities.
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Important Idea
2 4
2
b b acx
a
2 0ax bx c for
The quadratic formula:
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Try ThisSolve for principal values in radians:
2sec 2 tan 3x x
x=1.220 and x= -.632
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Important IdeaWhen solving trig
identities, try one or more of the following:•Factor if possible•Use quadratic formula if you can’t factor•Use identities to write the equation in terms of only one trig function.
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Lesson Close
Trignometric equations are used in science and engineering.