Solving Trig Equations

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Solving Trig Equations
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1 step problems solutions A B 2 step problems
solutions A B C 3 step problems solutions A B
C Multiple solutions solutions A B Overview
Trig Equations
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1 step problems
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Solving Trig Equations (A) (all in degrees, 0 x
360)
1) Solve Sin X 0.24
2) Solve Cos X 0.44
3) Solve Tan X 0.84
4) Solve Sin X -0.34
5) Solve Cos X -0.77
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1 step solns A
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1) Solve Sin x 0.24
Positive Sin so quadrant 1 2
x Sin-1 0.24
1st solution x 13.9
2nd solution x 180 13.9 166.1
Positive Cos so quadrant 1 4
2) Solve Cos X 0.44
x Cos-1 0.44
1st solution x 63.9
2nd solution x 360 63.9 296.1
3) Solve Tan X 0.84
Positive Tan so quadrant 1 3
x Tan-1 0.84
1st solution x 40.0
2nd solution x 180 40.0 220.0
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1 step solns B
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4) Solve Sin x -0.34
Negative Sin so quadrant 3 4
x Sin-1 0.34 19.9º
Use positive value
1st solution x 180 19.9 199.9º
2nd solution x 360 19.9 340.1
5) Solve Cos x -0.77
Negative Cos so quadrant 2 3
x Cos-1 0.77 39.6º
Use positive value
1st solution x 180 39.6 140.4º
2nd solution x 180 39.6 219.6
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2 step problems
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1) Solve 4Sin x 2.6
2) Solve Cos x 3 3.28
3) Solve 2Tan x 2 5.34
4) Solve 2 Sin x 1.85
5) Solve 0.5Cos x 3 2.6
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2 step solutions A
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1) Solve 4Sin x 2.6
Positive Sin so quadrant 1 2
Sin x 0.65
Divide by 4
x Sin-1 0.65 40.5º
1st solution x 40.5º
2nd solution x 180 40.5 139.5
2) Solve Cos x 3 3.28
Positive Cos so quadrant 1 4
Cos x 0.28
Subtract 3
x Cos-1 0.28 73.7º
1st solution x 73.7º
2nd solution x 360 73.7 286.3
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2 step solutions B
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3) Solve 2Tan x 2 5.34
Positive Tan so quadrant 1 3
Subtract 2
2Tan x 3.34
Divide by 2
Tan x 1.67
Inverse Tan
x Tan-1 1.67 59.1º
1st solution x 59.1º
2nd solution x 180 59.1 239.1
4) Solve 2 Sin x 1.85
Negative Sin so quadrant 3 4
Subtract 2
Sin x -0.15
x Sin-1 0.15 8.6º
Positive value
1st solution x 180 8.6 188.6º
2nd solution x 360 8.6 351.4
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2 step solutions C
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5) Solve 0.5Cos x 3 2.6
Negative Cos so quadrant 2 3
Subtract 3
0.5Cos x -0.4
Cos x -0.8
Divide by 0.5
x Cos-1 0.8 36.9º
Positive value
1st solution x 180 36.9 143.1
2nd solution x 180 36.9 216.9
The original graph y 0.5Cosx 3
y 2.6
x 143.9º
216.9º
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3 step problems
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1) Solve 2Sin(x 25) 1.5
2) Solve 5Cos(x 33) 4.8
3) Solve 2Tan(x 25) 8.34
4) Solve 5 Sin(x 45) 4.85
5) Solve 0.5Cos(x 32) 4 3.85
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3 Step solutions A
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1) Solve 2Sin(x 25) 1.5 Sin(x
25) 0.75 x 25 Sin-1 0.75 48.6 so x
23.6 x 25 180 48.6 131.4 so x
106.4
2) Solve 5Cos(x 33) 4.8 Let A x 33
so 5Cos(A) 4.8
Cos(A) 0.96 And x A 33 A Cos-1
0.96 16.3 or A 360 16.3 343.7 So x
16.3 33 -16.7 or x 343.7 33
310.7 But we need 2 solutions between 0 and
360! Next highest solution for A is A 360
16.3 376.3 So x 376.3 33 343.3 (or
-16.7 360) Solutions x 310.7 and
343.3
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3 step solutions B
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3) Solve 2Tan(x 25) 8.34 Tan(x 25)
4.17 x 25 Tan-1 4.17 76.5 so x
101.5 x 25 180 59.1 256.5 so x
281.5
4) Solve 5 Sin(x 45) 4.25 Sin(x
45) - 0.75 (Positive value) Sin-1 0.75
48.6 x 45 180 48.6 228.6 so x
183.6 x 45 360 48.6 311.4 so x
266.4
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3 step solutions C
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5) Solve 0.5Cos(x 32) 4 3.85
0.5Cos(x 32) -0.15 Cos(x 32)
-0.3 Cos-10.3 72.5 x
32 107.5 so x 75.5 OR x 32 252.5 so
x 220.5
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Multiple Solution Problems
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1) Solve Sin(2X) 0.6 2) Solve Cos(2X)
0.8 3) Solve 5Tan(2X) 8.4 4) Solve
Sin(2X 15) 0.85 5) Solve 0.5Cos(0.5X) 4
3.92
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Multiple solutions A
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1) Solve Sin(2x) 0.6 Let A 2x Sin(A)
0.6 A Sin-1 0.6 36.9
and A 180 36.9 143.1 The
next two solutions for A 396.9 and A
503.1 So A 36.9, 143.1,
396.9, 503.1 x A 2 so x 18.5 and
71.7 and 198.5 and 251.6
2) Solve Cos(2x) 0.8
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Multiple solutions B
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3) Solve 5Tan(2x) 8.4
4) Solve Sin(2x 15) 0.85
5) Solve 0.5Cos(0.5x) 4 3.92
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Overview Trig Equations
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1) Rearrange the equation into the form
Sin A
eg) Solve 5Sin(2px) 4
Sin(2px) 0.8
2) Find a solution to the trig equation
Check if degrees or radians!
Where A 2px
Sin(A) 0.8
A Sin-1 0.8 0.927 radians
3) Find several solutions for A
Using graph or unit circle
Note p so use radians
A p 0.927 2.214 rad
4) Use A to find solutions for x
A 0.927 or 2.214
Use each A to find x
x 0.927 2p 0.148
Where A 2px so x A 2p
x 2.214 2p 0.352