How to find the Trigonometrical Ratios of 0°?

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 small.

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

Now according to the definition of trigonometric ratio we get,

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 decreases and finally tends to zero then,

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

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

Hence, in the Limit when θ → 00 then \(\overline{PQ}\) → 0 and \(\overline{OP}\) → \(\overline{OQ}\) . Therefore, we get

\(\lim_{θ \to 0} sin θ

= \lim_{θ \rightarrow 0}\frac{\overline{PQ}}{\overline{OP}}

= \frac{0}{\overline{OQ}} \) [since, θ → 0° therefore, \(\overline{PQ}\) → 0].

= 0

Therefore** sin 0° = 0**

\(\lim_{θ \rightarrow 0} cos θ

= \lim_{θ \rightarrow 0}\frac{\overline{OQ}}{\overline{OP}}

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

= 1

Therefore** cos 0° = 1**

\(\lim_{θ \rightarrow 0} tan θ

= \lim_{θ \rightarrow 0}\frac{\overline{PQ}}{\overline{OQ}}

= \frac{0}{\overline{OQ}} \) [since, θ → 0° therefore, \(\overline{PQ}\) → 0].

= 0

Therefore** tan 0° = 0**

Thus,

csc 0° = \(\frac{1}{sin 0°}

= \frac{1}{0} \), [since, sin 0° = 0]

= undefined

Therefore** csc 0° = ****undefined**

sec 0° = \(\frac{1}{cos 0°}

= \frac{1}{1} \), [since, cos 0° = 1]

= 1** **

Therefore** sec 0° = 1**

cot 0° = \(\frac{1}{tan 0°}

= \frac{1}{0} \), [since, tan 0° = 0]

= undefined

Therefore** cot 0° = ****undefined**

Trigonometrical Ratios of 0 degree are commonly called standard angles and the trigonometrical ratios of these angles are frequently used to solve particular angles.

**●** **Trigonometric Functions**

**Basic Trigonometric Ratios and Their Names****Restrictions of Trigonometrical Ratios****Reciprocal Relations of Trigonometric Ratios****Quotient Relations of Trigonometric Ratios****Limit of Trigonometric Ratios****Trigonometrical Identity****Problems on Trigonometric Identities****Elimination of Trigonometric Ratios****Eliminate Theta between the equations****Problems on Eliminate Theta****Trig Ratio Problems****Proving Trigonometric Ratios****Trig Ratios Proving Problems****Verify Trigonometric Identities****Trigonometrical Ratios of 0°****Trigonometrical Ratios of 30°****Trigonometrical Ratios of 45°****Trigonometrical Ratios of 60°****Trigonometrical Ratios of 90°****Trigonometrical Ratios Table****Problems on Trigonometric Ratio of Standard Angle****Trigonometrical Ratios of Complementary Angles****Rules of Trigonometric Signs****Signs of Trigonometrical Ratios****All Sin Tan Cos Rule****Trigonometrical Ratios of (- θ)****Trigonometrical Ratios of (90° + θ)****Trigonometrical Ratios of (90° - θ)****Trigonometrical Ratios of (180° + θ)****Trigonometrical Ratios of (180° - θ)****Trigonometrical Ratios of (270° + θ)****Trigonometrical Ratios of (270° - θ)****Trigonometrical Ratios of (360° + θ)****Trigonometrical Ratios of (360° - θ)****Trigonometrical Ratios of any Angle****Trigonometrical Ratios of some Particular Angles****Trigonometric Ratios of an Angle****Trigonometric Functions of any Angles****Problems on Trigonometric Ratios of an Angle****Problems on Signs of Trigonometrical Ratios**

**11 and 12 Grade Math**

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