Transcript Solving Trig Equations

Solving Trig Equations
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
End
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1 step problems
Solving Trig Equations (A) (all in degrees, 0 ≤ x ≤ 360)
1) Solve Sin X = 0.24
180
S A
T C
2) Solve Cos X = 0.44
3) Solve Tan X = 0.84
0, 360
180
180
S A
T C
180
5) Solve Cos X = -0.77
180
T C
T C
0, 360
0, 360
4) Solve Sin X = -0.34
S A
S A
S A
T C
0, 360
0, 360
End
1 step solns A
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1) Solve Sin x = 0.24
Positive Sin so quadrant 1y& 2
1
x = Sin-1 0.24
1st solution: x = 13.9°
180
2nd solution: x = 180 – 13.9 = 166.1°
x=
1st
2nd
S A
T
1
1–
– 1
1
C
180
360
0, 360
180
360 x
– 1
Positive Cos so quadrant 1 & 4
2) Solve Cos X = 0.44
Cos-1
End
1
0.44
solution: x = 63.9°
180
solution: x = 360 – 63.9 = 296.1°
3) Solve Tan X = 0.84
S A
T C
0, 360
180
1
3
2
1
2
1–
2
3
– 3
1
2
3
180
360
360 x
– 1
Positive Tan so quadrant 1 & 3
x = Tan-1 0.84
1st solution: x = 40.0°
y
180
2nd solution: x = 180 + 40.0 = 220.0°
S A
T C
3
2
1
0, 360
– 1
– 2
– 3
y
180
360 x
1 step solns B
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4) Solve Sin x = -0.34
x = Sin-1 0.34 = 19.9º
End
1
1–
– 1
1
180
360
Negative Sin so quadrant 3 & 4
Use positive value
1st solution: x = 180 + 19.9° = 199.9º
1
180
y
2nd solution: x = 360 – 19.9 = 340.1°
180
S A
T C
0, 360
360 x
– 1
5) Solve Cos x = -0.77
x = Cos-1 0.77 = 39.6º
Negative Cos so quadrant 2 & 3
1–
– 1
1
1
180
360
Use positive value
180
1st solution: x = 180 – 39.6° = 140.4º
1
y
2nd solution: x = 180 + 39.6 = 219.6°
180
– 1
360 x
S A
T C
0, 360
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2 step problems
1) Solve 4Sin x = 2.6
180
S A
T C
0, 360
2) Solve Cos x + 3 = 3.28
180
3) Solve 2Tan x + 2 = 5.34
180
S A
T C
S A
T C
0, 360
0, 360
4) Solve 2 + Sin x = 1.85
180
S A
T C
5) Solve 0.5Cos x + 3 = 2.6
180
S A
T C
0, 360
0, 360
1
1–
– 1
1
180
360
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2 step solutions A
1) Solve 4Sin x = 2.6
Divide
by 4
End
Positive Sin so quadrant 1 & 2
Sin x = 0.65
1
y
x = Sin-1 0.65 = 40.5º
180
1st solution: x = 40.5º
360 x
180
S A
T C
0, 360
– 1
2nd solution: x = 180 – 40.5 = 139.5°
1–
– 1
1
1
180
360
2) Solve Cos x + 3 = 3.28
Positive Cos so quadrant 1 & 4
Cos x = 0.28
Subtract
3
1
y
x = Cos-1 0.28 = 73.7º
1st solution: x = 73.7º
180
– 1
2nd solution: x = 360 – 73.7 = 286.3°
360 x
180
S A
T C
0, 360
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2 step solutions B
3) Solve 2Tan x + 2 = 5.34
Subtract 2
2Tan x = 3.34
Divide by 2
Inverse Tan
1st
End
1
3
2
1
2
1–
2
3
– 3
1
2
3
180
360
Positive Tan so quadrant 1 & 3
Tan x = 1.67
x = Tan-1 1.67 = 59.1º
3
2
1
– 1
– 2
– 3
solution: x = 59.1º
y
180
180
360 x
S A
T C
0, 360
2nd solution: x = 180 + 59.1 = 239.1°
1
1–
– 1
1
180
360
4) Solve 2 + Sin x = 1.85
Subtract 2
Positive value
Negative Sin so quadrant 3 & 4
Sin x = -0.15
x = Sin-1 0.15 = 8.6º 1 y
180
1st solution: x = 180 + 8.6 = 188.6º
– 1
2nd solution: x = 360 – 8.6 = 351.4°
180
360 x
S A
T C
0, 360
1–
– 1
1
1
180
360
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End
2 step solutions C
5) Solve 0.5Cos x + 3 = 2.6
Subtract 3
0.5Cos x = -0.4
Divide by 0.5
Positive value
Negative Cos so quadrant 2 & 3
1
y
Cos x = -0.8
x=
Cos-1
0.8 =
180
4
3
2
1
360
1
2
3
4
180
360
36.9º
180
360 x
180
– 1
S A
T C
0, 360
1st solution: x = 180 – 36.9 = 143.1 °
2nd solution: x = 180 + 36.9 = 216.9°
y
4
3
The original graph
y = 0.5Cosx + 3
y = 2.6
2
1
180
x = 143.9º & 216.9º
360 x
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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
180
180
180
180
180
S A
T C
S A
T C
S A
T C
S A
T C
S A
T C
0, 360
0, 360
0, 360
0, 360
0, 360
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3 Step solutions A
End
1) Solve 2Sin(x + 25) = 1.5
S A
0, 360
180
Sin(x + 25) = 0.75
T C
-1
x + 25 = Sin 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
180
S A
T C
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°
0, 360
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End
3 step solutions B
3) Solve 2Tan(x – 25) = 8.34
180
S A
T C
0, 360
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
180
S A
T C
(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°
0, 360
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3 step solutions C
5) Solve 0.5Cos(x + 32) + 4 = 3.85
180
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°
S A
T C
0, 360
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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
End
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Multiple solutions A
End
1) Solve Sin(2x) = 0.6
Let A = 2x
Sin(A) = 0.6
S A
0, 360
180
A = Sin-1 0.6 = 36.9°
T C
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
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
180
180
180
S A
T C
S A
T C
S A
T C
0, 360
0, 360
0, 360
Overview: Trig Equations
Home
1) Rearrange the equation into the form
eg) Solve 5Sin(2πx) = 4
Sin A =
2) Find a solution to the trig equation
Check if degrees or radians!
3) Find several solutions for ‘A’
Using graph or unit circle
4) Use ‘A’ to find solutions for ‘x’
y
5
4
3
2
1
–
–
–
–
–
1
2
3
4
5
Sin(2πx) = 0.8
Where A = 2πx
Sin(A) = 0.8
A = Sin-1 0.8 = 0.927 radians
Note π so use radians
A = π – 0.927 = 2.214 rad
A = 0.927
or
2.214
Use each ‘A’ to find ‘x’
0.5
1 x
Where A = 2πx
so x = A ÷ 2π
x = 0.927 ÷ 2π = 0.148
x = 2.214 ÷ 2π = 0.352