If 3 sinθ + 4 cosθ = 5 and 0 degrees < θ < 90 degrees, then what is the value of sinθ?

Introduction / Context:

This trigonometry question tests your skill in manipulating linear combinations of sinθ and cosθ. Recognizing patterns involving Pythagorean triples can greatly simplify the problem. This is a very common style in entrance and aptitude tests.

Given Data / Assumptions:

• 3 sinθ + 4 cosθ = 5.
• The range of θ is 0 degrees < θ < 90 degrees, so sinθ and cosθ are positive.
• We must find sinθ.

Concept / Approach:

Notice the coefficients 3 and 4 and the right side 5. These resemble the 3, 4, 5 Pythagorean triple, where 3^2 + 4^2 = 5^2. This suggests that we can interpret sinθ and cosθ as legs of a right triangle and use this triple structure. One neat way is to suppose sinθ = 3/5 and cosθ = 4/5, then check whether the equation holds.

Step-by-Step Solution:

Verification / Alternative check:

Alternatively, you can write 3 sinθ + 4 cosθ in the form R sin(θ + φ), using R = square root of (3^2 + 4^2) = 5 and an angle φ such that cosφ = 3/5 and sinφ = 4/5. Then 3 sinθ + 4 cosθ = 5 sin(θ + φ) = 5, giving sin(θ + φ) = 1, so θ + φ = 90 degrees and sinθ evaluates to 3/5 again.

Why Other Options Are Wrong:

The values 1/5 and 2/5 are too small to satisfy the equation with any matching cosθ value. The value 4/5 would make cosθ smaller than 3/5 to preserve sin^2θ + cos^2θ = 1, which breaks the given linear relation. The value 1 would force cosθ = 0, which does not satisfy 3 sinθ + 4 cosθ = 5.

Common Pitfalls:

Learners may try to square both sides and create a more complicated equation, or they may forget that both sinθ and cosθ must be positive in the first quadrant. Recognizing the 3, 4, 5 pattern quickly leads to an elegant solution.

Final Answer:

The value of sinθ is 3/5.