1. Solving Trig Equations
End
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

2. Home
1 step problems
End
Solving Trig Equations (A) (all in degrees, 0 ≤ x ≤ 360)
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0, 360 A T C S
1) Solve Sin X = 0.24
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2) Solve Cos X = 0.44
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3) Solve Tan X = 0.84
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4) Solve Sin X = -0.34
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5) Solve Cos X = -0.77

3. 1 step solns A Home End 1) Solve Sin x = 0.24
Positive Sin so quadrant 1 & 2
x = Sin
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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
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0, 360 A T C S
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
180
0, 360 A T C S
1st solution: x = 40.0°
2nd solution: x = = 220.0°

4. 1 step solns B Home End 4) Solve Sin x = -0.34
Negative Sin so quadrant 3 & 4
x = Sin = 19.9º
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0, 360 A T C S
Use positive value
1st solution: x = ° = 199.9º
2nd solution: x = 360 – 19.9 = 340.1°
5) Solve Cos x = -0.77
Negative Cos so quadrant 2 & 3
x = Cos = 39.6º
Use positive value
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1st solution: x = 180 – 39.6° = 140.4º
2nd solution: x = = 219.6°

5. 2 step problems Home End 1) Solve 4Sin x = 2.6
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2) Solve Cos x + 3 = 3.28
3) Solve 2Tan x + 2 = 5.34
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4) Solve 2 + Sin x = 1.85
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5) Solve 0.5Cos x + 3 = 2.6
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6. 2 step solutions A Home End 1) Solve 4Sin x = 2.6
Positive Sin so quadrant 1 & 2
Sin x = 0.65
Divide by 4
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x = Sin = 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 = 73.7º
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1st solution: x = 73.7º
2nd solution: x = 360 – 73.7 = 286.3°

7. 2 step solutions B Home End 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
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Inverse Tan
x = Tan = 59.1º
1st solution: x = 59.1º
2nd solution: x = = 239.1°
4) Solve 2 + Sin x = 1.85
Negative Sin so quadrant 3 & 4
Subtract 2
Sin x = -0.15
x = Sin = 8.6º
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Positive value
1st solution: x = = 188.6º
2nd solution: x = 360 – 8.6 = 351.4°

8. 2 step solutions C Home End 5) Solve 0.5Cos x + 3 = 2.6
Negative Cos so quadrant 2 & 3
Subtract 3
0.5Cos x = -0.4
Divide by 0.5
Cos x = -0.8
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x = Cos = 36.9º
Positive value
1st solution: x = 180 – 36.9 = °
2nd solution: x = = 216.9°
The original graph
y = 0.5Cosx + 3
y = 2.6
x = 143.9º
& 216.9º

9. 3 step problems Home End 1) Solve 2Sin(x + 25) = 1.5
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1) Solve 2Sin(x + 25) = 1.5
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2) Solve 5Cos(x + 33) = 4.8
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3) Solve 2Tan(x – 25) = 8.34
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4) Solve 5 + Sin(x + 45) = 4.85
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5) Solve 0.5Cos(x + 32) + 4 = 3.85

10. 3 Step solutions A 1) Solve 2Sin(x + 25) = 1.5 Sin(x + 25) = 0.75
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3 Step solutions A
End
1) Solve 2Sin(x + 25) = 1.5
Sin(x + 25) = 0.75
x + 25 = Sin = 48.6° so x = 23.6°
x + 25 = 180 – 48.6 = 131.4° so x = 106.4°
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2) Solve 5Cos(x + 33) = 4.8
Let A = x so 5Cos(A) = 4.8
Cos(A) = 0.96
And x = A – 33
A = Cos = 16.3° or A = 360 – 16.3 = 343.7
So x = 16.3 – 33 = -16.7° or x = – 33 = 310.7°
But we need 2 solutions between 0 and 360!
Next highest solution for A is A = = 376.3°
So x = 376.3° – 33 = 343.3° (or -16.7° + 360)
Solutions: x = 310.7° and °
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11. 3 step solutions B 3) Solve 2Tan(x – 25) = 8.34 Tan(x – 25) = 4.17
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3 step solutions B
End
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3) Solve 2Tan(x – 25) = 8.34
Tan(x – 25) = 4.17
x – 25 = Tan = 76.5° so x = 101.5°
x – 25 = =256.5° so x = 281.5°
4) Solve 5 + Sin(x + 45) = 4.25
Sin(x + 45) =
(Positive value) Sin = 48.6°
x + 45 = = 228.6° so x = 183.6°
x + 45 = 360 – 48.6 = 311.4° so x = 266.4°
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12. 3 step solutions C 5) Solve 0.5Cos(x + 32) + 4 = 3.85
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3 step solutions C
End
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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13. Multiple Solution Problems
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Multiple Solution Problems
End
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

14. Multiple solutions A 1) Solve Sin(2x) = 0.6 Let A = 2x Sin(A) = 0.6
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Multiple solutions A
End
1) Solve Sin(2x) = 0.6
Let A = 2x Sin(A) = 0.6
A = Sin = 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°, °, °, °
x = A ÷ 2 so x = 18.5° and 71.7° and 198.5° and 251.6°
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2) Solve Cos(2x) = 0.8

15. Multiple solutions B Home End 3) Solve 5Tan(2x) = 8.4
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3) Solve 5Tan(2x) = 8.4
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4) Solve Sin(2x + 15) = 0.85
180
0, 360 A T C S
5) Solve 0.5Cos(0.5x) + 4 = 3.92

16. Overview: Trig Equations
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Overview: Trig Equations
1) Rearrange the equation into the form
Sin A =
eg) Solve 5Sin(2πx) = 4
Sin(2πx) = 0.8
2) Find a solution to the trig equation
Check if degrees or radians!
Where A = 2πx
Sin(A) = 0.8
A = Sin = radians
3) Find several solutions for ‘A’
Using graph or unit circle
Note π so use radians
A = π – = rad
4) Use ‘A’ to find solutions for ‘x’
A = or
Use each ‘A’ to find ‘x’
Where A = 2πx so x = A ÷ 2π
x = ÷ 2π = 0.148
x = ÷ 2π = 0.352