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8/8/2019 Trigonometric Equation 1
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TRIGONOMETRIC
EQUATION
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10/7/2010 lingeswari~ 2
There is no general method in solvingtrigonometric equations.
The following examples shows the solution ofdifferent types of trigonometric equations.
1) Simple trigonometric equations.2) Quadratic equations.
3) Equations involving compound and double-angle-formulae.
4) Equations involving factor formulae.
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5) Equations in the form of a sin b cos = c where a,b,care constants.these equations can be solved by using thefollowing transformations:
a sin + b cos = r sin ( + )
a sin b cos = r sin ( ) a cos + b sin = r cos ( ) a cos + b sin = r cos ( + )
where a > 0 , b > 0
r = a + btan = b a0< < 90
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A.SIMPLE TRIGONOMETRIC EQUATION
1) Solve the following equations for 0 < < 360
i. cot = 3
tan = 1/ 3
= 30, ( 180 + 30 ) = 30, 210
ii. cot 2 = 1/2
2 = 45, ( 360- 45) , ( 360 + 45 ) , (720 45 )2 = 45, 315, 405, 675
= 22 , 157 , 202 , 337
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2) Solve the following sinx = -0.4
* Since sinx is negative,x lies in the third / fourth quadrant.
Case 1: assume that x lies in the third quadrantx = 180 +
sinx = sin ( 180 + )
= -sin
-sin = -0.4
sin = 0.4
= 23.58
x
= 180 +23.5
8= 203.58
Since sin ( 2n +x ) = sinx = -0.4, we get a set ofanswers,namely
2n + 203.5
8 , n Z.
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Case 2 : assume that x lies in the fourth quadrant.
x= 360
sinx= ( 360 )
= - sin -sin = -0.4
sin = 0.4
= 23.58
x = 360 23.58
= 336.42
Since sin ( 2n +x) = sinx= -0.4, we get a set of answers namely2n + 336.42, n Z.
From cases 1 and 2,we see that the answers to the equation sinx= -0.4 are
x= 2n + 203.58 , 2n + 336.42 , n Z. Notice that in either case 1 or case 2, we are
actually solving sin = 0.4 where is an acute angle. Then letx= 180 + 0 or
x= 360 , ex: + or 2 0. Then,we write the general solutions as :x= 2n + ( + ) or x= 2n + ( 2 )
= (2n +1) + , n Z. = 2 (n + 1) , n Z.
Usually,in questions,xwill be defined. For example, you maybe asked to solve the
equation sinx= -0.4 forxlying in the interval [ 0,2 ]. In this case , the answers are
x= 203.58 andx= 336.42
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ii. 3tan 5 sec + 1 = 0
3(sec 1) 5 sec + 1 = 03 sec 5 sec 2 = 0
3(sec 1) (sec 2) = 0
sec = -1/3 @sec = 2
cos = -3 no solutioncos =
= 60 , 300
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2) Solve the equation 2cos t + sin t = 1, 0 t 2
2 cos t + sin t = 12( 1 - sin t) + sin t = 12 sin t sin t 1 = 0(2 sin t + 1) (sin t 1) = 0
sin t = -1/2 @ sin t = 1
If sin t = 1, then t = 90. Note that we only need to
find t lying in the interval [0, 2]. If sin t = -1/2,then lies either in third or fourth quadrant
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Case 1: Assume t lies in third quadrant
t = 180 +
sin t = sin (180 + sin )= -sin sin =
= 30t = 210
Case 2: Assume t lies in the fourth quadrant
t = 360 sin t = sin ( 360 )
= -sin
sin = = 30t = 330
The answers to the equation 2 cos t + sin t = 1 are 90, 210,330