Solve 5 cos(3 x) = 2 for the smallest three positive solutions.
Give your answers accurate to at least two decimal places, as a list separated by commas
To solve the equation 5 cos(3x) = 2 for the smallest three positive solutions, we need to isolate the variable x.
Step 1: Divide both sides of the equation by 5:
cos(3x) = 2/5
Step 2: Take the inverse cosine (cos^(-1)) of both sides of the equation:
3x = cos^(-1)(2/5)
Step 3: Divide both sides of the equation by 3:
x = (1/3) * cos^(-1)(2/5)
Now, we can use a calculator to find the value of cos^(-1)(2/5) accurate to at least two decimal places.
Using a calculator:
cos^(-1)(2/5) ≈ 1.159
Finally, substitute this value back into the equation for x:
x = (1/3) * 1.159
x ≈ 0.386
Therefore, the first positive solution for 5 cos(3x) = 2 is x ≈ 0.386.
To find the next two solutions, we need to consider the periodicity of the cosine function. The cosine function has a period of 2π, meaning that it repeats every 2π radians or 360 degrees.
To find the second solution, we add one period (2π) to the value we found for x:
x = 0.386 + 2π
x ≈ 6.069
For the third solution, we add two periods (4π) to the initial value:
x = 0.386 + 4π
x ≈ 12.753
So, the smallest three positive solutions to 5 cos(3x) = 2 accurate to at least two decimal places are x ≈ 0.386, 6.069, and 12.753.
To solve the equation 5cos(3x) = 2, we need to isolate the variable x.
Step 1: Divide both sides of the equation by 5:
cos(3x) = 2/5
Step 2: Take the inverse cosine (cos^(-1)) of both sides to eliminate the cosine function:
3x = cos^(-1)(2/5)
Step 3: Simplify and isolate x:
x = (1/3) * cos^(-1)(2/5)
Using a calculator, we find:
x ≈ 0.722, 1.222, 1.763
Therefore, the smallest three positive solutions to the equation 5cos(3x) = 2 are approximately:
x ≈ 0.72, 1.22, 1.76