2-3cosx=5+3cosx 0 degrees is less than or equal to x less than or equal to 180 degrees
Can you please show me step by step in detail. Please do not skip a step. Thanks!
add 3 cos x to each side
2=5 + 6 cos x
6cosx=-3
cosx=-1/2
x=arccos (-.5)and you can figure it from there
120 degrees?
To solve the equation 2 - 3cos(x) = 5 + 3cos(x) for the given restriction that 0 degrees is less than or equal to x less than or equal to 180 degrees, follow these steps:
Step 1: Combine like terms:
Start by adding 3cos(x) to both sides of the equation:
2 - 3cos(x) + 3cos(x) = 5 + 3cos(x) + 3cos(x)
Simplifying this equation gives:
2 = 5 + 6cos(x)
Step 2: Move constants to one side:
Next, subtract 5 from both sides of the equation:
2 - 5 = 5 + 6cos(x) - 5
Simplifying this equation gives:
-3 = 6cos(x)
Step 3: Isolate cos(x):
Divide both sides of the equation by 6:
-3/6 = 6cos(x)/6
Simplifying this equation gives:
-1/2 = cos(x)
Step 4: Determine the angle:
Now, we need to find the angle x where the cosine function equals -1/2.
In the given range of 0 degrees to 180 degrees, the angles where cos(x) = -1/2 are 120 degrees and 240 degrees.
Step 5: Verify the solution:
To verify if these angles indeed satisfy the original equation, substitute each value back into the equation and check if both sides are equal:
When x = 120 degrees:
2 - 3cos(120) = 5 + 3cos(120)
Simplifying both sides gives:
2 + (3/2) = 5 - (3/2)
(4/2) + (3/2) = (8/2) - (3/2)
7/2 = 5/2
When x = 240 degrees:
2 - 3cos(240) = 5 + 3cos(240)
Simplifying both sides gives:
2 + (3/2) = 5 - (3/2)
(4/2) + (3/2) = (8/2) - (3/2)
7/2 = 5/2
Both angles satisfy the equation, so the solutions are x = 120 degrees and x = 240 degrees.