Concept:In a right triangle, trigonometric ratios are defined by side lengths. tanθ = opposite/adjacent, and secθ = hypotenuse/adjacent. Use the Pythagorean theorem to find the missing side.
Explanation:Given tanθ = 3/4. This means opposite side : adjacent side = 3 : 4. Let opposite = 3k and adjacent = 4k, where k is a positive constant.
Apply Pythagoras: hypotenuse² = (3k)² + (4k)² = 9k² + 16k² = 25k². Thus hypotenuse = 5k.
Now secθ = hypotenuse / adjacent = (5k) / (4k) = 5/4. The constant k cancels, giving secθ = 5/4.
Alternatively, use the identity sec²θ = 1 + tan²θ. Substituting tanθ = 3/4 gives sec²θ = 1 + 9/16 = 25/16, so secθ = 5/4 (positive as θ is acute). Both methods confirm the same value.
Answer:45 (Option A)