If tan(A) = 3⁄5, then cos(A) is equal to:

Correct — D. The stem is printed as an image: tan(A) = 3⁄5. In a right triangle tan = opposite ⁄ adjacent, so take the opposite side as 3 and the adjacent side as 5.

Hypotenuse by Pythagoras:
h² = 3² + 5² = 9 + 25 = 34
h = √34

cos A = adjacent ⁄ hypotenuse = 5⁄√34 → option (d), the picture showing 5 over √34.

Why the others are wrong

• (a)
  The picture shows 3⁄√34, which is sin A — the opposite side over the hypotenuse. Cosine takes the adjacent side, 5, over the same √34.
• (b)
  9⁄√34 squares the 3 (3² = 9) while leaving the hypotenuse as it is. The squares only belong in the Pythagoras step, never in the ratio itself.
• (c)
  25⁄√34 squares the adjacent side (5² = 25). cos A needs the length 5, and 25⁄√34 is larger than 1, which no cosine can be.

A single trigonometric ratio fixes the shape of the right triangle, so every other ratio follows.

tan A = 3⁄5 means the two legs are in the ratio 3 : 5, so you may call them 3k and 5k. Pythagoras then gives the hypotenuse √34 k, and the k cancels out of every ratio you form.

That is why the answer is a fixed number even though the triangle's actual size is never given.

Key facts

• For tan A = p⁄q the legs are p and q and the hypotenuse is √(p² + q²).
• tan A = 3⁄5 gives hypotenuse √34, so sin A = 3⁄√34 and cos A = 5⁄√34.
• sin²A + cos²A = 1 checks the pair: 9⁄34 + 25⁄34 = 34⁄34 = 1.

Common traps

• Reading 3⁄5 as the sine ratio and answering 3⁄√34
• Squaring a leg in the ratio because √34 came from squares
• Choosing a value greater than 1 for a sine or cosine

No directly related past PYQ was found.