# Trigonometrical Ratios of 90°

How to Find the Trigonometrical Ratios of 90°?

Let a rotating line  $$\overrightarrow{OX}$$ rotates about O in the anti-clockwise sense and starting from its initial position $$\overrightarrow{OX}$$ traces out ∠XOY = θ where θ is very nearly equal to 90°.

Let $$\overrightarrow{OX}$$ ⊥ $$\overrightarrow{OZ}$$ therefore, ∠XOZ = 90°

Take a point P on $$\overrightarrow{OY}$$ and draw $$\overline{PQ}$$ perpendicular to $$\overline{OX}$$.

Then,

Sin θ = $$\frac{\overline{PQ}}{\overline{OP}}$$;

cos θ = $$\frac{\overline{OQ}}{\overline{OP}}$$

and tan θ =$$\frac{\overline{PQ}}{\overline{OQ}}$$

When θ is slowly approaches 90° and finally tends to 90° then,

(a) $$\overline{OQ}$$ slowly decreases and finally tends to zero and

(b) the numerical difference between $$\overline{OP}$$  and $$\overline{PQ}$$  becomes very small and finally tends to zero.

Hence, in the Limit when θ → 90° then $$\overline{OQ}$$ → 0 and $$\overline{PQ}$$   → $$\overline{OP}$$  . Therefore, we get

$$\lim_{θ \rightarrow 90°}$$ sin  θ

= $$\lim_{θ \rightarrow 90°}\frac{\overline{PQ}}{\overline{OP}}$$

= $$\frac{\overline{OP}}{\overline{OP}}$$ [since, θ → 90° therefore, $$\overline{PQ}$$   → $$\overline{OP}$$ ].

= 1

Therefore sin 90° = 1

$$\lim_{θ \rightarrow 90°}$$ cos θ

= $$\lim_{θ \rightarrow 90°}\frac{\overline{OQ}}{\overline{OP}}$$

= $$\frac{0}{\overline{OP}}$$, [since, θ → 0° therefore, $$\overline{OQ}$$ → 0].

= 0

Therefore cos 90° = 0

$$\lim_{θ \rightarrow 90°}$$ tan θ

= $$\lim_{θ \rightarrow 90°}\frac{\overline{PQ}}{\overline{OQ}}$$

= $$\frac{\overline{OP}}{0}$$ [since, θ → 0° $$\overline{OQ}$$ → 0 and $$\overline{PQ}$$   → $$\overline{OP}$$].

= undefined

Therefore tan 900 = undefined

Thus,

csc 90° = $$\frac{1}{sin 90°}$$

= $$\frac{1}{1}$$, [since, sin 90° = 1]

= 1

sec 90° = $$\frac{1}{cos 90°}$$

= $$\frac{1}{0}$$, [since, cos  90° = 0]

= undefined

cot 0° = $$\frac{ cos 90°}{ sin 90°}$$

= $$\frac{0}{1}$$, [since, sin 900 = 1 and cos 90° = 0]

= 0

Trigonometrical Ratios of 90 degree are commonly called standard angles and the trigonometrical ratios of these angles are frequently used to solve particular angles.

Trigonometric Functions