Trigonometrical Ratios of 30°

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°.

Trigonometrical Ratios of 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, OA2 + PA2 = OP2

⇒ OA2 + a2 = (2a)2

⇒ OA2 = 4a2 – a2

⇒ OA2 = 3a2

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.

 Trigonometric Functions








11 and 12 Grade Math

From Trigonometrical Ratios of 30° to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Addition and Subtraction of Fractions | Solved Examples | Worksheet

    Jul 18, 24 03:08 PM

    Addition and subtraction of fractions are discussed here with examples. To add or subtract two or more fractions, proceed as under: (i) Convert the mixed fractions (if any.) or natural numbers

    Read More

  2. Worksheet on Simplification | Simplify Expressions | BODMAS Questions

    Jul 18, 24 01:19 AM

    In worksheet on simplification, the questions are based in order to simplify expressions involving more than one bracket by using the steps of removal of brackets. This exercise sheet

    Read More

  3. Fractions in Descending Order |Arranging Fractions an Descending Order

    Jul 18, 24 01:15 AM

    We will discuss here how to arrange the fractions in descending order. Solved examples for arranging in descending order: 1. Arrange the following fractions 5/6, 7/10, 11/20 in descending order. First…

    Read More

  4. Fractions in Ascending Order | Arranging Fractions | Worksheet |Answer

    Jul 18, 24 01:02 AM

    Comparison Fractions
    We will discuss here how to arrange the fractions in ascending order. Solved examples for arranging in ascending order: 1. Arrange the following fractions 5/6, 8/9, 2/3 in ascending order. First we fi…

    Read More

  5. Worksheet on Comparison of Like Fractions | Greater & Smaller Fraction

    Jul 18, 24 12:45 AM

    Worksheet on Comparison of Like Fractions
    In worksheet on comparison of like fractions, all grade students can practice the questions on comparison of like fractions. This exercise sheet on comparison of like fractions can be practiced

    Read More