How to find the trigonometrical Ratios of 30°?

Let a rotating line \(\overrightarrow{OX}\) rotates about O in the anti-clockwise sense and starting from the initial position \(\overrightarrow{OX}\) traces out ∠XOY = 30°.

Take a point P on \(\overrightarrow{OY}\) and draw PA perpendicular to \(\overrightarrow{OX}\) Then, ∠OPA = 60°.

Now, produce PA to B such that PA = MB and join OB.From ∆PMO and ∆QMO we have,

PA = BA,

OA common

and ∠OBP = ∠OPB = 60°

Therefore, ∠POB = 30° + 30° = 60°; which shows that each angel of triangle OPQ is 60° . Hence ∆OPQ is equilateral.

Let, OP = PB = 2a; therefore, PA = ½ PB = a

Again, OA

⇒ OA

⇒ OA

⇒ OA

Therefore, OA = √3a (Since, OA > 0).

Now, from the right-angled ∆OPA we have,

sin 30° = \(\frac{\overline{PA}}{\overline{OP}} = \frac{a}{2a} = \frac{1}{2}\);

cos 30° = \(\frac{\overline{OA}}{\overline{OP}} = \frac{\sqrt{3}a}{2a} = \frac{\sqrt{3}}{2}\)

And tan 30° = \(\frac{PA}{OA} = \frac{a}{\sqrt{3}a} = \frac{1}{\sqrt3} = \frac{\sqrt{3}}{3}\)

Therefore, csc 30° = \(\frac{1}{sin 30°}\) = 2;

Sec 30° = \(\frac{1}{cos 30°} = \frac{2}{\sqrt3} = \frac{2\sqrt{3}}{3}\)

And cot 30° = \(\frac{1}{tan 30°}\) = √3.

Trigonometrical Ratios of 30° are commonly called standard angles and the trigonometrical ratios of these angles are frequently used to solve particular angles.

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**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**

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