Solving Equations Containing Circular Functions :
Solve 2cos sq x =1
To solve the equation 2cos^2x = 1, we need to isolate the cosine function and find the value(s) of x that satisfy the equation.
Step 1: Rewrite the equation in terms of the inverse cosine function (arccos):
cos^2x = 1/2
Step 2: Take the square root of both sides of the equation:
cosx = ±√(1/2)
Step 3: Simplify the square root:
cosx = ±√(1)/√(2)
cosx = ±1/√(2)
Step 4: Rationalize the denominator by multiplying the numerator and denominator by √(2):
cosx = ±(1/√(2)) * (√(2)/√(2))
cosx = ±√(2)/2
Now, we have two possible values for cosine: √(2)/2 and -√(2)/2.
Step 5: Find the angles that have these cosine values.
Using the unit circle or reference angles, we can determine the following solutions:
x = π/4 + 2πn, or x = 7π/4 + 2πn
where n is any integer.
These are the solutions to the equation 2cos^2x = 1.