The solutions in the interval [0, 2π] for the equation
are approximately is x ≈ 0.469 radians and x ≈ 2.672 radians.
To solve the trigonometric equation
in the interval [0, 2π], you can follow these steps:
Step 1: Rewrite the equation to simplify it:

Multiply both sides of the equation by 3 to eliminate the fraction:

Step 2: Use the trigonometric identity
= 1 to replace
:

Step 3: Expand and simplify the equation:

Step 4: Combine like terms:

Step 5: Solve for
:

Divide both sides by -10:

Step 6: Take the square root of both sides (remember to consider both positive and negative square roots):
cos(x) = ±√(9/10)
Step 7: Find the values of x in the interval [0, 2π] where cos(x) is equal to ±√(9/10).
First, find the values where cos(x) = √(9/10):
x = arccos(√(9/10))
Using a calculator to find the approximate value:
x ≈ 0.469 radians
Now, find the values where cos(x) = -√(9/10):
x = arccos(-√(9/10))
Using a calculator to find the approximate value:
x ≈ 2.672 radians
So, The answer is approximately x ≈ 0.469 radians and x ≈ 2.672 radians.