Solve for x if 3sin2x - 3sinx = 2 - 4cosx 0<x<180
6sinxcosx - 3sinx = 2 - 4cosx
3sinx(2cosx - 1) = 2(1 - 2cosx)
3sinx(2cosx - 1) - 2(1 - 2cosx) = 0
3sinx(2cosx - 1) + 2(2cosx - 1)
(2cosx - 1)(3sinx + 2) = 0
cosx = 1/2
x = 60° or 300°
or
3sinx = -2
sinx = -2/3, so x is in III or IV
from my calculator, sin 41.81° = +2/3
x = 180+41.81 = 221.81°
or
x = 360-41.81 = 318.19°
To solve the equation 3sin2x - 3sinx = 2 - 4cosx, we can start by simplifying the equation using trigonometric identities.
Step 1: Rewrite sin2x using the double-angle identity.
sin2x = 2sinxcosx
Substituting this into the original equation, we have:
3(2sinxcosx) - 3sinx = 2 - 4cosx
Simplifying further:
6sinxcosx - 3sinx = 2 - 4cosx
Step 2: Rearrange the equation to one side:
6sinxcosx - 3sinx + 4cosx = 2
Step 3: Factor out sinx from the first two terms and cosx from the third term:
sinx(6cosx - 3) + 4cosx = 2
Step 4: Combine the terms with sinx:
(6cosx - 3)sinx + 4cosx = 2
Step 5: Divide both sides by (6cosx - 3):
sinx = (2 - 4cosx)/(6cosx - 3)
Step 6: Simplify the right side further:
sinx = (2(1 - 2cosx))/(3(2cosx - 1))
Step 7: Use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to eliminate sinx:
1 - cos^2(x) = (2(1 - 2cosx))/(3(2cosx - 1))
Step 8: Multiply both sides by 3(2cosx - 1):
3(2cosx - 1) - 3cos^2(x) = 2(1 - 2cosx)
Step 9: Distribute and rearrange terms:
6cosx - 3 - 3cos^2(x) = 2 - 4cosx
Step 10: Combine like terms:
3cos^2(x) + 10cosx - 1 = 0
Step 11: Solve the quadratic equation. Let's use the quadratic formula:
cosx = (-b ± √(b^2 - 4ac)) / (2a)
In this case:
a = 3, b = 10, c = -1
cosx = (-10 ± √(10^2 - 4(3)(-1))) / (2(3))
cosx = (-10 ± √(100 + 12)) / 6
cosx = (-10 ± √112) / 6
cosx = (-10 ± 4√7) / 6
Step 12: Simplify the solutions:
cosx = (-5 ± 2√7) / 3
Step 13: Since 0 < x < 180, we know that cosx cannot be negative. Therefore, we take the positive solution:
cosx = (-5 + 2√7) / 3
Step 14: Solve for x by taking the inverse cosine:
x = cos^(-1)((-5 + 2√7) / 3)
Approximately, x ≈ 24.79 degrees.
To solve the equation 3sin2x - 3sinx = 2 - 4cosx, we can apply trigonometric identities and simplification techniques. Here's how to solve it step by step:
1. Start with the equation: 3sin2x - 3sinx = 2 - 4cosx
2. Convert sin2x to a double angle formula: sin2x = 2sinxcosx
Rewrite the equation as: 3(2sinxcosx) - 3sinx = 2 - 4cosx
3. Distribute: 6sinxcosx - 3sinx = 2 - 4cosx
4. Combine like terms: 6sinxcosx - 3sinx + 4cosx = 2
5. Rearrange terms: 6sinxcosx - 3sinx + 4cosx - 2 = 0
6. Group the terms that contain sinx: (6sinxcosx - 3sinx) + (4cosx - 2) = 0
7. Factor out sinx from the first group: sinx(6cosx - 3) + (4cosx - 2) = 0
8. Factor out cosx from the second group: sinx(6cosx - 3) + cosx(4 - 2) = 0
9. Combine like terms: sinx(6cosx - 3) + 2cosx = 0
10. Factor out (6cosx - 3): (6cosx - 3)(sinx + 2) = 0
Now we have two equations:
1. 6cosx - 3 = 0
2. sinx + 2 = 0
Solving equation 1:
6cosx - 3 = 0
Add 3 to both sides: 6cosx = 3
Divide by 6: cosx = 1/2
Applying inverse cosine: x = arccos(1/2)
x = 60 degrees or x = 300 degrees (within the given range 0 < x < 180)
Solving equation 2:
sinx + 2 = 0
Subtract 2 from both sides: sinx = -2
Since -2 is outside the range [-1, 1], there are no values of x that satisfy this equation.
Therefore, the solution for x in the given range is x = 60 degrees or x = 300 degrees.