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

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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

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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

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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

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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

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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