For an acute angle θ greater than 0 that satisfies the trigonometric equation cos^2θ - 3cosθ + 2 = sin^2θ, what is the value of θ in degrees?

Introduction / Context:
This problem checks understanding of basic trigonometric identities and algebraic manipulation. We are given an equation that involves both cos^2θ and sin^2θ, and we must determine which acute angle θ satisfies it.

Given Data / Assumptions:

• θ is an acute angle, so 0° < θ < 90°.
• The equation cos^2θ - 3cosθ + 2 = sin^2θ holds.
• We must find θ in degrees from the given options.

Concept / Approach:
We use the fundamental identity sin^2θ + cos^2θ = 1 to rewrite sin^2θ in terms of cos^2θ. This converts the equation into a quadratic equation in cosθ. Then we solve that quadratic, check which solutions lie in the acute range, and finally read off θ from standard trigonometric values.

Step-by-Step Solution:
Step 1: Replace sin^2θ using sin^2θ = 1 - cos^2θ. Step 2: The equation becomes cos^2θ - 3cosθ + 2 = 1 - cos^2θ. Step 3: Move all terms to one side: cos^2θ - 3cosθ + 2 - 1 + cos^2θ = 0, so 2cos^2θ - 3cosθ + 1 = 0. Step 4: Factor the quadratic: 2cos^2θ - 3cosθ + 1 = (2cosθ - 1)(cosθ - 1) = 0. Step 5: So cosθ = 1 or cosθ = 1/2. Step 6: For θ acute, cosθ = 1 gives θ = 0°, which is not allowed since θ > 0°. Step 7: For cosθ = 1/2, the acute angle is θ = 60°.

Verification / Alternative check:
Substitute θ = 60° directly into the original equation. We have cos60° = 1/2, so cos^2 60° = 1/4. The left side becomes 1/4 - 3(1/2) + 2 = 1/4 - 3/2 + 2 = 3/4. On the right side, sin60° = √3/2, so sin^2 60° = 3/4. Both sides match, confirming θ = 60° is correct.

Why Other Options Are Wrong:
Option A (90°) gives cos90° = 0 and sin^2 90° = 1, which does not satisfy the quadratic relation. Option B (30°) gives cosθ = √3/2, and the equation fails numerically. Option C (45°) leads to cosθ = √2/2 and also does not satisfy the equation. Only 60° makes both sides equal.

Common Pitfalls:
Learners sometimes forget to apply sin^2θ + cos^2θ = 1 correctly, or they treat cosθ as θ and solve a wrong quadratic. Another mistake is to accept θ = 0° without checking the acute condition. Always verify that your angle lies in the specified interval and that it satisfies the original equation, not just the transformed one.

Final Answer:

The only acute angle that satisfies the given equation is 60°.