Let θ be an acute angle (0 < θ < π/2) such that 5 sin^2(θ) + 4 cos^2(θ) = 9/2. Using this relation, determine the exact value of tan(θ).

Introduction / Context:
This trigonometry question combines the Pythagoras identity sin^2(θ) + cos^2(θ) = 1 with a given linear combination of sin^2(θ) and cos^2(θ). The goal is to find tan(θ), which is the ratio sin(θ) / cos(θ). Such problems are common in aptitude tests and exam papers, because they check your ability to use basic identities creatively and to solve for trigonometric ratios from transformed equations.

Given Data / Assumptions:

• θ is an acute angle, so 0 < θ < π/2 and all primary trigonometric ratios are positive.
• The relation is 5 sin^2(θ) + 4 cos^2(θ) = 9/2.
• We must find tan(θ) in simplest exact form.

Concept / Approach:
Introduce temporary variables s = sin^2(θ) and c = cos^2(θ). We know that s + c = 1 from the Pythagoras identity. The given equation 5s + 4c = 9/2 can be combined with s + c = 1 to solve for s and c individually. Once sin^2(θ) and cos^2(θ) are known, we can take positive square roots (since θ is acute) to find sin(θ) and cos(θ), and then compute tan(θ) = sin(θ) / cos(θ).

Step-by-Step Solution:
1) Let s = sin^2(θ) and c = cos^2(θ). 2) From the basic identity, s + c = 1. 3) The given equation is 5s + 4c = 9/2. 4) Express c in terms of s using s + c = 1: c = 1 − s. 5) Substitute c into 5s + 4c = 9/2 to get 5s + 4(1 − s) = 9/2. 6) Simplify: 5s + 4 − 4s = s + 4. 7) So s + 4 = 9/2. Subtract 4 from both sides: s = 9/2 − 4 = 9/2 − 8/2 = 1/2. 8) Therefore sin^2(θ) = 1/2. Since θ is acute, sin(θ) = 1/√2. 9) Using s + c = 1 again, c = 1 − s = 1 − 1/2 = 1/2, so cos^2(θ) = 1/2 and cos(θ) = 1/√2. 10) Now compute tan(θ) = sin(θ) / cos(θ) = (1/√2) / (1/√2) = 1.

Verification / Alternative check:
We can verify by substituting sin^2(θ) = 1/2 and cos^2(θ) = 1/2 back into the original equation. The left side becomes 5 * (1/2) + 4 * (1/2) = 5/2 + 4/2 = 9/2, which matches the right side exactly. This confirms that sin^2(θ) and cos^2(θ) are correct. With both squares equal, sin(θ) and cos(θ) have the same positive value for an acute angle, so tan(θ) = 1 is consistent with the standard angle θ = 45 degrees, which is a well known solution where sin(θ) = cos(θ).

Why Other Options Are Wrong:
Options b (1/4), d (1/2), and c (2) correspond to tan^2(θ) values that would make sin^2(θ) and cos^2(θ) unequal in a way that does not satisfy the given linear combination. Option e (0) would imply sin(θ) = 0, which cannot satisfy 5 sin^2(θ) + 4 cos^2(θ) = 9/2. Only tan(θ) = 1 leads to sin^2(θ) and cos^2(θ) both equal to 1/2, which fits the equation perfectly.

Common Pitfalls:
A typical mistake is to try to manipulate tan(θ) directly without first solving for sin^2(θ) and cos^2(θ). Another error is substituting c = 1 − s incorrectly, leading to algebraic mistakes such as sign errors or incorrect simplification of 9/2 − 4. Some learners also forget that θ is acute and might consider negative square roots. Carefully using the identity s + c = 1 and remembering that trigonometric ratios are positive in the first quadrant avoids these problems.

Final Answer:
The value of tan(θ) that satisfies the given relation is 1.