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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. Solving Trig Equations. A. A. A. A. A. S. S. S. S. S. 0, 360. 0, 360. 0, 360. 0, 360. 0, 360. 180. 180.

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Solving Trig Equations

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  1. End 1 step problems: solutions AB 2 step problems: solutions ABC 3 step problems: solutions ABC Multiple solutions: solutions AB Overview: Trig Equations Solving Trig Equations

  2. A A A A A S S S S S 0, 360 0, 360 0, 360 0, 360 0, 360 180 180 180 180 180 T T T T T C C C C C Home End 1 step problems 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

  3. A A A S S S 0, 360 0, 360 0, 360 180 180 180 T T T C C C Home End 1 step solns A 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°

  4. A A S S 0, 360 0, 360 180 180 T T C C Home End 1 step solns B 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°

  5. A A A A A S S S S S 0, 360 0, 360 0, 360 0, 360 0, 360 180 180 180 180 180 T T T T T C C C C C Home End 2 step problems 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

  6. A A S S 0, 360 0, 360 180 180 T T C C Home End 2 step solutions A 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°

  7. A A S S 0, 360 0, 360 180 180 T T C C Home End 2 step solutions B 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°

  8. A S 0, 360 180 T C Home End 2 step solutions C 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 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

  9. A A A A A S S S S S 0, 360 0, 360 0, 360 0, 360 0, 360 180 180 180 180 180 T T T T T C C C C C Home End 3 step problems 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

  10. A A S S 0, 360 0, 360 180 180 T T C C Home End 3 Step solutions A 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°

  11. A A S S 0, 360 0, 360 180 180 T T C C Home End 3 step solutions B 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°

  12. A S 0, 360 180 T C Home End 3 step solutions C 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°

  13. Home End Multiple Solution Problems 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. A S 0, 360 180 T C Home End Multiple solutions A 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

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

  16. Home 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-1 0.8 = 0.927 radians 3) Find several solutions for ‘A’ Using graph or unit circle Note πso use radians A = π – 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÷2π= 0.148 Where A = 2πx so x = A ÷ 2π x = 2.214÷ 2π= 0.352

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