section 7.3 trigonometric equations
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Section 7.3
Trigonometric Equations
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Determine whether is a solution of the equation 2sin 2 0. 4
5Is a solution?4
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22sin 2 2 2 2 2 2 2 04 2
is NOT a solution.4
22sin 2 2 2 2 24
052
5 is a solution.4
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2cos 3 0
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3cos2
3Where is cos ? On the interval 2
0,2 there are two answers, one in
quadrant I and one in quadrant IV.
11On 0, 2 , or =6 6
A general formula would include all coterminal angles112 or = 2
6 6k k
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Solve the equation: 2cos 2 1 0, 0 2
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1cos 22
1On the interval 0, 2 the cosine of an angle is 2
5when the angle is or .3 3
52 or 2 but since we divide by 2, adding 2 to the angle 3 3
will still give us a solution for that falls in the interval 0,2 .
5 52 , 2 +2 , 2 , 2 23 3 3 3
5 7 11, , , 6 6 6 6
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Solve the equation: 3 tan 3 1 0, 0 2
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1tan 33
On the interval 0,2 the tangent of an angle is
3 5 11when the angle is or .3 6 6
5 113 or 3 but since we divide by 3, adding 2 and 4 to 6 6
the angle will still give us a solution for that falls in the interval 0,2 .
5 5 5 11 11 113 , 3 +2 , 3 +4 ,3 , 3 2 ,3 46 6 6 6 6 6
5 11 17 23 29 35, , , , , 18 18 18 18 18 18
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Solve the equation: cos 1, 0 24
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The cosine equals 1 when the angle is 0.
04
4
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cos 0.2
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1cos 0.2 1.37 For inverse cosine the calculator will only give an angle from 0 to π. There is another angle in the interval from 0 to 2π with that cosine value in quadrant IV.
x
y
1.37 2 1.37
1.37,4.91
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2Solve the equation: 2cos cos 1 0, 0 2
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2cos 1 cos 1 0
2cos 1 0 or cos 1 0
1cos or cos 12
2 4 , , 03 3
2 40, ,3 3
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2 2Solve the equation: sin sin cos , 0 2
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2 2sin sin 1 sin 22sin sin 1 0
2sin 1 sin 1 0
2sin 1 0 or sin 1 0
1sin or sin 12
7 11 , ,6 6 2
7 11, ,2 6 6
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2 2Solve the equation: 3sin cos , 0 2
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2 23sin 1 sin 24sin 1 2 1sin4
1 1sin4 2
6
,56
,76
,11
6
6
,56
, 76
,116