Arrange sin 30°, cos 45° and tan 30° in increasing order of value.

Class 10Introduction to TrigonometryTrigonometry

Asin 30° < tan 30° < cos 45°
Btan 30° < sin 30° < cos 45°
Csin 30° < cos 45° < tan 30°
Dcos 45° < tan 30° < sin 30°

Step-by-Step Solution

sin 30° =

\frac{1}{2} = 0.50

, tan 30° =

\frac{1}{3}

≈ 0.58 and cos 45° =

\frac{1}{2}

≈ 0.71. So sin 30° < tan 30° < cos 45°.

Answer: A) sin 30° < tan 30° < cos 45°

Key idea: Why ratios — and not lengths

Two right triangles with the same acute angle θ are SIMILAR (Class 10 ch. 6 — AA criterion: both have a right angle and the angle θ).

Similar triangles → corresponding sides in fixed ratio.
So the ratio (opposite / hypotenuse) takes the SAME value in any right triangle with angle θ — regardless of how big the triangle is.

That's why sin θ, cos θ, tan θ are FUNCTIONS of the angle alone — they're well-defined.

This is the fundamental reason trigonometry works as a clean theory.

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