Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval [0,2π)[0,2𝜋). Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."2cos(x)−2⎯⎯√=0
Question
Use trigonometric identities, algebraic methods, and inverse trigonometric functions, as necessary, to solve the following trigonometric equation on the interval [0,2π)[0,2𝜋). Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."2cos(x)−2⎯⎯√=0
Solution
The given equation is 2cos(x) - √2 = 0.
Step 1: Isolate the cosine function.
To do this, add √2 to both sides of the equation to get:
2cos(x) = √2
Step 2: Solve for cos(x).
Divide both sides of the equation by 2 to get:
cos(x) = √2 / 2
Step 3: Find the value of x.
Use the inverse cosine function to find the value of x.
x = cos^(-1)(√2 / 2)
The values of x that satisfy this equation on the interval [0, 2π) are x = π/4 and x = 7π/4.
So, the solutions to the equation are x = π/4 and x = 7π/4.
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