Trigonometric Equations : Session 1

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Trigonometric Equations Session 1
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Illustrative Problem
Solve sinx cosx 2
Solution
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Definition

• A trigonometric equation
• is an equation
• Contains trigonometric functions of variable
  angle

sin ? ½
2 sin2? sin22? 2.
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Periodicity and general solution
Solution of Trigonometric Equation Values of ?,
which satisfy the trigonometric equation
For sin ? ½ , ? ? /6, 5 ? /6, 13 ? /6,.
No. of solutions are infinite .
Why ? - Periodicity of trigonometric functions.
e.g. - sin?, cos ? have a period as 2 ?
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Periodicity and general solution
Periodicity of trigonometric functions.
f(?T) f(?)
sin? have a period 2 ?
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Graph of ysinx
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Graph of ycosx
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Graph of ytanx
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Periodicity and general solution
As solutions are infinite , the entire set of
solution can be written in a compact form.
This compact form is referred to as general
solution
For sin ? ½ , ? ?/6, 5?/6, 13?/6,.
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Principal Solutions
Solutions in 0?x?2?
? principal solutions.
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Illustrative problem
Find the principal solutions of tanx
Solution
? principal solutions are 5?/6 and 11?/6
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Illustrative problem
Find the principal solution of the equation sinx
1/2
Solution
sin?/6 1/2 and sin(?- ?/6) 1/2 ? principal
solution are x ?/6 and 5?/6.
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General solution of sin ? 0
sin? PM/OP For sin? 0 , PM 0
For PM 0, OP will lie on XOX
? ? 0, p, 2p, 3p ..
? ? is an integral multiple of p.
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General solution of sin ? 0
For sin ? 0 , ? is an integer multiple of p.
Or ? np , n ? Z (n belongs to set of integers)
Hence, general solution of sin? 0 is ? np ,
where n ? Z,
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General solution of cos ? 0
cos ? OM/OP For cos ? 0 , OM 0
For OM 0, OP will lie on YOY
? ? p/2, 3p/2, 5p/2.
? is an odd integer multiple of p/2.
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General solution of cos ? 0
For cos ? 0 , ? is an odd integer multiple
of p/2.
Or ? (2n1)p/2 , n ? Z (n belongs to set of
integers)
Hence, general solution of cos? 0 is ?
(2n1)p/2 , where n ? Z,
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General solution of tan ? 0
tan? PM/OM For tan? 0 , PM 0
For PM 0, OP will lie on XOX
? ? 0,p, 2p, 3p.
? is an integer multiple of p.
Same as sin ? 0
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General solution of tan ? 0
For tan ? 0 , ? is an integer multiple of p.
Or ? np , n ? Z (n belongs to set of integers)
Hence, general solution of tan? 0 is ? np,
where n ? Z,
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Illustrative Problem
Find the general value of x satisfying the
equation sin5x 0
Solution
sin5x 0 sin0
gt 5x n?
gt x n?/5
gtx n?/5 where n is an integer
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General solution of sin ? k
If sin? k ? -1? k ? 1
Let k sin?, choose value of ? between ?/2 to ?
/2
If sin? sin? ? sin? - sin? 0
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General solution of sin ? k
? 2np ?
? (2n1)p - ?
Even , ve
Odd , -ve
? np (-1)n? , where n ? Z
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General solution of cos ? k
If cos? k ? -1? k ? 1
Let k cos?, choose value of ? between 0 to ?
If cos? cos? ? cos? - cos? 0
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General solution of cos ? k
? 2np ?
? 2np - ?
-ve
ve
? 2np ? ? , where n ? Z
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General solution of tan ? k
If tan? k ? - ? lt k lt ?
Let k tan?, choose value of ? between - ?/2 to
?/2
If tan? tan? ? tan? - tan? 0
? sin?.cos? - cos?.sin? 0
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General solution of tan ? k
sin?.cos? - cos?.sin? 0

• sin( ? - ? ) 0
• ? - ? np , where n ? Z
• ? ? np ?

? np ? , where n ? Z
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Illustrative problem
Find the solution of sinx
Solution
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Illustrative problem
Solve tan2x
Solution
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Illustrative problem
Solve sin2x sin4x sin6x 0
Solution
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Illustrative Problem
Solve 2cos2x 3sinx 0
Solution
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General solution of sin2x sin2? cos2x cos2?,
tan2x tan2?
n? ? ? where n is an integer.
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Illustrative Problem
Solve 4cos3x-cosx 0
Solution
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Illustrative Problem
Solve sinx siny 2
Solution
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Class Exercise Q1.
Solve sin5x cos2x
Solution
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Class Exercise Q2.
Solve 2sinx 3cosx5
Solution
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Class Exercise Q3.
Solve 7cos2? 3sin2? 4
Solution
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Class Exercise Q4.
Solution
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Class Exercise Q5.
Show that 2cos2(x/2)sin2x x2x-2 for 0ltxlt?/2
has no real solution.
Solution
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Class Exercise Q6.
Solution
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Class Exercise Q6.
Solution
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Class Exercise Q7.
Solve the equation sinx cosx 1sinxcosx
Solution
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Class Exercise Q7.
Solve the equation sinx cosx 1sinxcosx
Solution
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Class Exercise Q8.
Solution
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Class Exercise Q8.
If rsinx3,r4(1sinx), then x
is
Solution
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Class Exercise Q9.
Solution
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Class Exercise Q10.
Solve the equation (1-tan?)(1sin2?) 1tan?
Solution
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Class Exercise Q10.
Solve the equation (1-tan?)(1sin2?) 1tan?
Solution
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Class Exercise Q10.
Solve the equation (1-tan?)(1sin2?) 1tan?
Solution