Let 0° < t < 90°. Then which of the followings is true?

Correct — C. All four choices are pictures in the response sheet. Option (c) reads sin(t) < cos(t) when t < 45°.

Rule: cos t = sin(90° − t), so the comparison becomes a comparison of two angles.

For 0° < t < 45°, the partner angle 90° − t lies between 45° and 90°.
So 90° − t is the larger angle.

sin is increasing across 0° to 90°, so the larger angle has the larger sine:
sin(90° − t) > sin t, i.e. cos t > sin t → option (c).

Check at t = 30°: sin 30° = 0.5, cos 30° ≈ 0.866.

Why the others are wrong

• (a)
  Above 45° the inequality flips. For 45° < t < 90° the partner angle 90° − t is the smaller one, so sin t > cos t — at t = 60°, sin is 0.866 against cos 0.5.
• (b)
  At exactly 45° the two are equal, not unequal: sin 45° = cos 45° = 1⁄√2 ≈ 0.707. That is the single crossing point on this interval, so the ≠ makes the claim false.
• (d)
  This reverses the true order below 45°. Sine is the smaller of the pair there — sin 30° = 0.5 against cos 30° ≈ 0.866 — so sin t > cos t fails for every t under 45°.

Across 0° < t < 90°, sin climbs from 0 to 1 while cos falls from 1 to 0, so the two curves cross exactly once.

That crossing is at 45°, where sin 45° = cos 45° = 1⁄√2. Every part of this item follows from that one point: below 45° cosine is larger, above it sine is larger.

The proof needs no calculator. cos t = sin(90° − t) converts the question into which of t and 90° − t is bigger, and sine being increasing settles it.

Key facts

• cos t = sin(90° − t) for every angle t, which turns a sine-versus-cosine comparison into an angle comparison.
• sin 45° = cos 45° = 1⁄√2 ≈ 0.707, the only angle between 0° and 90° where the two are equal.
• On 0° to 90° sine is increasing and cosine is decreasing.
• tan t = 1 exactly at t = 45°, the same crossover expressed a third way.

Common traps

• Testing only t = 45°, where the strict inequalities in three of the options say nothing.
• Reasoning that sine grows and is therefore always the larger of the pair.
• Skimming options (a) and (c) as the same statement because both carry the < sign.

No directly related past PYQ was found.