solve the following eq. [0<=x<=360]
2cos(3x-72)=1
Well I haven't done trig in a long time but u believe...
You divide the 2 to each side to get
Cos(3x-72)=1/2
Then do the inverse to both sides to get rid of the cos and end with 3x-72=60 degrees
So x= 44 degrees
Cosine is positive in quad1&4
So 44degrees and 316degrees.
44&124degrees
but 3x mean it has turned 360 degree three times, how about the other four ans?
I plugged in 3x-72=___ (set them equal to 420,660,780,1020)
Answer:44,124,164,244,284 degrees
2cos(3x-72) = 1
cos(x-7) = 1/2
we know that cos 60° = 1/2 and cos 300° = 1/2
( (inverse)cos .5) = 60 )
3x-72 = 60 or 3x-72=300
3x = 132 or 3x = 372
x = 44° or x = 124° ---> your basic angles
but the period of cos(3x-72) is 360/3 = 120°
so as long as we don't go beyond your given domain, we can add 120° to any previous answer to get a new answer, so
x = 44+120 = 164
x = 164+120 = 284
x = 284+120 = 404 ---> too big
x = 124+120 = 244
x = 244+120364 -- > too big
so x = 44°, 124°, 164°, 244°, and 284°
just like L.Bianchessi found for you in the above post
To solve the equation 2cos(3x-72) = 1, we will follow these steps:
Step 1: Rewrite the equation
2cos(3x-72) = 1
Step 2: Bring the cosine term to one side
2cos(3x-72) - 1 = 0
Step 3: Apply the inverse function
To solve this equation, we need to apply the inverse function of cosine, which is the arc cosine (also known as acos or cos^(-1)). This will help us isolate the variable x.
acos(2cos(3x-72) - 1) = acos(0)
Step 4: Solve for x
Apply the inverse function of cosine to the right side of the equation as well:
3x - 72 = acos(0)
Now, we have:
3x - 72 = arccos(0)
Step 5: Evaluate arccos(0)
The arccos function evaluates the angle whose cosine is equal to 0. In this case, the cosine is equal to 0 at two specific angles: 90 degrees (π/2 radians) and 270 degrees (3π/2 radians).
Therefore, we have two equations to solve separately:
1) 3x - 72 = 90
2) 3x - 72 = 270
Solving equation 1):
3x - 72 = 90
Add 72 to both sides:
3x = 162
Divide by 3:
x = 54
Solving equation 2):
3x - 72 = 270
Add 72 to both sides:
3x = 342
Divide by 3:
x = 114
Step 6: Check the solutions
To ensure the solutions are within the given range, which is 0 ≤ x ≤ 360, we need to check whether x = 54 and x = 114 satisfy this condition.
For x = 54:
Since 54 is within the range from 0 to 360, this solution is valid.
For x = 114:
Since 114 is within the range from 0 to 360, this solution is valid as well.
Therefore, the solutions to the equation 2cos(3x-72) = 1, within the given range, are x = 54 and x = 114.