Determine the values of x that satisfy cos^2(x) - 1/2 = 0.

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Question: Determine the values of x that satisfy cos^2(x) - 1/2 = 0.

Options:

Correct Answer: π/4, 3π/4

Solution:

The solutions are x = π/4 and x = 3π/4.

Determine the values of x that satisfy cos^2(x) - 1/2 = 0.

Determine the values of x that satisfy cos^2(x) - 1/2 = 0.

Correct Answer: x = π/4 and x = 3π/4

Step 1: Start with the equation cos^2(x) - 1/2 = 0.
Step 2: Add 1/2 to both sides of the equation to isolate cos^2(x).
Step 3: The equation now looks like cos^2(x) = 1/2.
Step 4: Take the square root of both sides. Remember to consider both the positive and negative roots: cos(x) = ±√(1/2).
Step 5: Simplify √(1/2) to 1/√2 or √2/2.
Step 6: Now we have two equations: cos(x) = √2/2 and cos(x) = -√2/2.
Step 7: Find the angles x that satisfy cos(x) = √2/2. These angles are x = π/4 and x = 7π/4 (but we will focus on the first two quadrants for this problem).
Step 8: Find the angles x that satisfy cos(x) = -√2/2. These angles are x = 3π/4 and x = 5π/4 (but we will focus on the first two quadrants for this problem).
Step 9: The solutions in the range [0, 2π) are x = π/4 and x = 3π/4.

Trigonometric Identities – Understanding and applying the identity cos^2(x) + sin^2(x) = 1 to solve for x.
Quadratic Equations – Recognizing the equation as a quadratic in terms of cos(x) and solving for its roots.
Unit Circle – Using the unit circle to find angles corresponding to specific cosine values.