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.



Share this page: What’s this?

Recent Articles

  1. Method of L.C.M. | Finding L.C.M. | Smallest Common Multiple | Common

    Apr 15, 24 01:29 AM

    LCM of 24 and 30
    We will discuss here about the method of l.c.m. (least common multiple). Let us consider the numbers 8, 12 and 16. Multiples of 8 are → 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, ......

    Read More

  2. Common Multiples | How to Find Common Multiples of Two Numbers?

    Apr 15, 24 01:13 AM

    Common multiples of two or more given numbers are the numbers which can exactly be divided by each of the given numbers. Consider the following. (i) Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24…

    Read More

  3. Least Common Multiple |Lowest Common Multiple|Smallest Common Multiple

    Apr 14, 24 03:06 PM

    Lowest Common Multiple
    The least common multiple (L.C.M.) of two or more numbers is the smallest number which can be exactly divided by each of the given number. The lowest common multiple or LCM of two or more numbers is t…

    Read More

  4. Worksheet on H.C.F. | Word Problems on H.C.F. | H.C.F. Worksheet | Ans

    Apr 14, 24 02:23 PM

    HCF Using Venn Diagram
    Practice the questions given in the worksheet on hcf (highest common factor) by factorization method, prime factorization method and division method. Find the common factors of the following numbers…

    Read More

  5. Common Factors | Find the Common Factor | Worksheet | Answer

    Apr 14, 24 02:01 PM

    Common Factors of 24 and 36
    Common factors of two or more numbers are a number which divides each of the given numbers exactly. For examples 1. Find the common factor of 6 and 8. Factor of 6 = 1, 2, 3 and 6. Factor

    Read More