Solving Trig Equations1 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
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
180 0, 360A
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Solving Trig Equations (A) (all in degrees, 0 ≤ x ≤ 360)
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1 step solns A1) Solve Sin x = 0.24
1st solution: x = 13.9° 180 0, 360A
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2) Solve Cos X = 0.44
3) Solve Tan X = 0.84
Positive Sin so quadrant 1 & 2
x = Sin-1 0.24
2nd solution: x = 180 – 13.9 = 166.1°
1st solution: x = 63.9° 180 0, 360A
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Positive Cos so quadrant 1 & 4
x = Cos-1 0.44
2nd solution: x = 360 – 63.9 = 296.1°
1st solution: x = 40.0° 180 0, 360
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Positive Tan so quadrant 1 & 3x = Tan-1 0.84
2nd solution: x = 180 + 40.0 = 220.0°
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1 step solns B4) Solve Sin x = -0.34
1st solution: x = 180 + 19.9° = 199.9º
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Negative Sin so quadrant 3 & 4
x = Sin-1 0.34 = 19.9º
2nd solution: x = 360 – 19.9 = 340.1°
Use positive value
5) Solve Cos x = -0.77
1st solution: x = 180 – 39.6° = 140.4º
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Negative Cos so quadrant 2 & 3
x = Cos-1 0.77 = 39.6º
2nd solution: x = 180 + 39.6 = 219.6°
Use positive value
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2 step problems
5) Solve 0.5Cos x + 3 = 2.6
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
180 0, 360A
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2 step solutions A EndHome
1) Solve 4Sin x = 2.6
1st solution: x = 40.5º 180 0, 360
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Positive Sin so quadrant 1 & 2
x = Sin-1 0.65 = 40.5º
2nd solution: x = 180 – 40.5 = 139.5°
Sin x = 0.65Divide by 4
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2) Solve Cos x + 3 = 3.28
1st solution: x = 73.7º 180 0, 360
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Positive Cos so quadrant 1 & 4
x = Cos-1 0.28 = 73.7º
2nd solution: x = 360 – 73.7 = 286.3°
Cos x = 0.28Subtract
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2 step solutions BHome
3) Solve 2Tan x + 2 = 5.34
1st solution: x = 59.1º
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Positive Tan so quadrant 1 & 3
x = Tan-1 1.67 = 59.1º
2nd solution: x = 180 + 59.1 = 239.1°
Tan x = 1.67Divide by 2
2Tan x = 3.34Subtract 2
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Inverse Tan
4) Solve 2 + Sin x = 1.85
1st solution: x = 180 + 8.6 = 188.6º180 0, 360
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Negative Sin so quadrant 3 & 4
x = Sin-1 0.15 = 8.6º
2nd solution: x = 360 – 8.6 = 351.4°
Sin x = -0.15Subtract 2
Positive value
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2 step solutions C5) Solve 0.5Cos x + 3 = 2.6
1st solution: x = 180 – 36.9 = 143.1 °
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Negative Cos so quadrant 2 & 3
x = Cos-1 0.8 = 36.9º
2nd solution: x = 180 + 36.9 = 216.9°
0.5Cos x = -0.4Subtract 3
Cos x = -0.8Divide by 0.5
Positive value
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The original graph
y = 0.5Cosx + 3
y = 2.6
x = 143.9º
180 360 1 2 3 4
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& 216.9º
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3 step problems
5) Solve 0.5Cos(x + 32) + 4 = 3.85
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
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3 Step solutions A
180 0, 360A
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S1) Solve 2Sin(x + 25) = 1.5
Sin(x + 25) = 0.75x + 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) = 0.96And x = A – 33 A = Cos-1 0.96 = 16.3° or A = 360 – 16.3 = 343.7So 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 B180 0, 360
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S3) 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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S5) 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 Problems1) 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
4) Solve Sin(2x + 15) = 0.85
3) Solve 5Tan(2x) = 8.4
5) Solve 0.5Cos(0.5x) + 4 = 3.92
180 0, 360A
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Overview: Trig EquationsHome
eg) Solve 5Sin(2πx) = 4
Sin(A) = 0.8
Sin(2πx) = 0.8
1) Rearrange the equation into the form Sin A =
Where A = 2πx2) Find a solution to the trig equation Check if degrees or radians!
Note π so use radians3) Find several solutions for ‘A’ Using graph or unit circle
A = π – 0.927 = 2.214 rad4) Use ‘A’ to find solutions for ‘x’
A = 0.927 or 2.214
x = 0.927 ÷ 2π = 0.148
x = 2.214 ÷ 2π = 0.352
A = Sin-1 0.8 = 0.927 radians
Use each ‘A’ to find ‘x’
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Where A = 2πx so x = A ÷ 2π