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As θ increases from 0° towards 90°, the value of tan θ:

Class 10Introduction to TrigonometryTrigonometry
  • Adecreases steadily from 1 to 0
  • Bincreases from 0 up to 1 and stops there
  • Cstays constant at 1 throughout
  • Dincreases from 0 and grows without any upper limit, and is undefined at 90°

Step-by-Step Solution

tan θ = . As θ grows, sin θ rises towards 1 while cos θ falls towards 0, so the ratio keeps getting larger (tan 45° = 1, tan 60° = ≈ 1.73, tan 89° ≈ 57). At 90° the denominator is 0, so tan 90° is undefined.
Answer: D) increases from 0 and grows without any upper limit, and is undefined at 90°

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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