If θ is a positive acute angle and 7 cos²θ + 3 sin²θ = 4, then what is the exact value of θ in degrees?

Introduction / Context:
This trigonometry question checks how well you can use the Pythagorean identity sin²θ + cos²θ = 1 and manipulate equations involving sin²θ and cos²θ. The key idea is to convert the expression into a single trigonometric function, solve for its value, and then identify the corresponding acute angle θ in degrees.

Given Data / Assumptions:

• θ is a positive acute angle (0° < θ < 90°).
• The equation 7 cos²θ + 3 sin²θ = 4 holds.
• Standard trigonometric identity sin²θ + cos²θ = 1 can be used.
• All trigonometric values are in degrees for this question.

Concept / Approach:
The main idea is to rewrite the equation so that it involves only one trigonometric function. Since sin²θ and cos²θ are linked by sin²θ = 1 − cos²θ, substituting this into the expression will convert everything into cos²θ. Then we can solve a simple algebraic equation in cos²θ, obtain cos θ, and match it with a standard angle whose cosine is known exactly.

Step-by-Step Solution:
Start with 7 cos²θ + 3 sin²θ = 4. Use sin²θ = 1 − cos²θ. So 7 cos²θ + 3(1 − cos²θ) = 4. Expand: 7 cos²θ + 3 − 3 cos²θ = 4. Combine like terms: (7 − 3) cos²θ + 3 = 4. Therefore 4 cos²θ + 3 = 4. So 4 cos²θ = 1. Hence cos²θ = 1/4. For an acute angle, cos θ is positive, so cos θ = 1/2. The acute angle with cos θ = 1/2 is θ = 60°.

Verification / Alternative check:
Substitute θ = 60° back into the original equation. We have cos 60° = 1/2, so cos² 60° = 1/4. Also sin 60° = √3/2, so sin² 60° = 3/4. Then 7 cos² 60° + 3 sin² 60° = 7(1/4) + 3(3/4) = 7/4 + 9/4 = 16/4 = 4, which exactly matches the right side of the equation, confirming that θ = 60° is correct.

Why Other Options Are Wrong:
30° gives cos² 30° = 3/4 and sin² 30° = 1/4, which leads to 7(3/4) + 3(1/4) = 24/4 = 6, not 4. 45° makes cos² 45° = sin² 45° = 1/2, giving 7/2 + 3/2 = 5, not 4. 90° is not even valid here because cos 90° = 0 which leads to 3, not 4. The option 75° does not produce the required balance either.

Common Pitfalls:
Common mistakes include forgetting to use sin²θ = 1 − cos²θ, mishandling the algebra when combining terms, or taking cos θ = −1/2 even though the angle is specified as acute. Remember that for acute angles, sine and cosine are both positive, so the positive square root must be chosen for cos θ in this context.

Final Answer:
Therefore, the required acute angle is 60°.