Sin 30 Degrees

Sin is a trigonometric ratio and the value of sin 30 degrees is half (½). Now the question arises what is a trigonometric ratio and how the value of sin 30 degrees is calculated. Trigonometric ratios is a part of trigonometry that deals with right-angled triangles. A triangle is a closed figure having three sides, three angles and three edges. If one of the three angles of a triangle is of 90 degree then it is called a right-angled triangle. The three sides of a right-angled triangle are as follows:


  1. Hypotenuse = It is the longest side of every right-angled triangle.


  1. Base = It has both angles that are 90-degree (right angle) and theta (unknown angle). It is also called adjacent. 


  1. Perpendicular = It does not have an unknown angle (theta). It is also called the opposite.


Trigonometric Ratios

There are six trigonometric ratios, namely, Sin, Cos, Tan, Cosec, Sec, and Cot which are actually the ratio of the sides of a right-angled triangle. 


Let us consider a triangle ∆ABC, in which ∠C = 90°. The side AB (opposite to the right angle) is always the hypotenuse because it is the longest side. So, the side AB named as c is the hypotenuse in this particular case. Side CB is base and side CA is perpendicular.


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Sinϴ = Perpendicular/ Hypotenuse


Cosϴ =  Base / Hypotenuse


Tanϴ = Perpendicular/ Base


Reciprocals:


  1. The reciprocal or inverse of Sin is Cosec. That is,

If Sinϴ = Perpendicular / Hypotenuse then Cosec ϴ = Hypotenuse /Perpendicular


  1. The reciprocal or inverse of Cos is Sec. That is,

If Cosϴ = Base / Hypotenuse then Sec ϴ = Hypotenuse/ Base


  1. The reciprocal or inverse of Tan is Cot. That is,

If Cosϴ = Base / Hypotenuse then Sec ϴ = Hypotenuse /Base



Usually, the trigonometric ratios are calculated for all the angles less than 90 degrees but given below are the basic ones. 

Basic degrees: 0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°. 


Given below is a for the values of all trigonometric ratios of the standard trigonometric angles, that is, 0°, 30°, 45°, 60°, and 90°. In this chapter, we are going to discuss the value of sin 30 degrees. 


Trigonometric Values

Angleϴ


Ratio

    0°

    30°

      45°

      60°

      90°

Sin ϴ

    0

    1/2

      1/√2

    √3/2

      1

Cos ϴ

    1

    √3/2

      1/√2

      1/2

      0

Tan ϴ

    0

    1/√3

      1

      √3

Not Defined

Cosec ϴ

Not Defined

    2

      √2

    2/√3

      1

Sec ϴ

    1

    2/√3

      √2

      2

Not Defined

Cot ϴ

Not Defined

      √3

      1

    1/√3

        0


According to this sin 30 table, the value of sin 30 degree is ½.

 

Sine of 30 Degrees Value

In order to express the sine function of an acute angle ϴ of a right-angled triangle ABC, it is important to name the sides based on the angles. The three sides of sin 30 triangle are given as follows:


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  • The longest side of the triangle that is, the side C is the hypotenuse. It is right opposite to the right-angled triangle and also contains the unknown angle theta.

  • The side B is considered as a base (adjacent) not only because triangle rests on it but also because it has both the angles, that is, 90 degree and unknown angle theta ϴ 

  • Side A is the perpendicular (opposite) as it is the only side that does not contain the angle ϴ and is adjacent to the base.


As we know, 

The sine function of an angle is equal to the ratio of the length of perpendicular to the length of Hypotenuse and the formula is given by,


Sinϴ = Perpendicular /Hypotenuse.


Sine Law:

The sine law affirms that “the sides of a triangle are proportional to the sine of the opposite angles.


Let us take a normal triangle ABC, 

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Now, according to the rule, 


a/Sin A = b/Sin B= c/Sin C = d


We use sine law when:

  1. Two angles and one side of a triangle given.

  2. Two sides and one included angles are given.


Derivation to Find the Sin 30 Value

Let us consider an equilateral triangle ABC having all the angles as 60 degrees. Now, the question is what is the value of sin 30  and what is the opposite of sin ?

Hence to find the answer of sin 30 value  we need to know the length of all the sides of the triangle.


So, let us suppose that AB=2a, such that half of each side is a.


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To find the value of sin 30 degree, we will use the following formula,


Sinϴ = Perpendicular Hypotenuse.


           Sin 30° = BD/ABa/2a = 12


Thus, the value of Sin 30 degree is equal to 12(half) or 0.5.


Just like the way we derived the value of sin 30 degrees, we can derive the value of sin degrees like 0°, 30°, 45°, 60°, 90°,180°, 270° and 360°. 

 

Solved Examples

Example 1: In triangle XYZ, right-angled at Y, XY = 10 cm and angle XZY = 30°. Find the length of the side XZ.


Solution:

To find the length of the side XZ, we use the formula of the sine function, which is ,

Sin 30°= Perpendicular Hypotenuse

Sin 30°= XY / XZ

On substituting the value of sin 30

½ = XY/ XZ

½ = 10/ XZ

XZ = 20cm 

 

Therefore, the length of the side, XZ = 20 cm.


Example 2: How do I find the value of sin(-30)?

Solution: 

Sin (-30) = - Sin (30)

Sin 30 = ½

Therefore sin (-30) = - ½ .

FAQ (Frequently Asked Questions)

Question 1: What are the trigonometric identities?

Answer: Trigonometric identities are equalities that include trigonometric functions and hold true for every value of the occurring variables such that both the sides of the equality are defined. These identities also involve certain functions of one or more angles. Trigonometric identities are useful whenever expressions demanding trigonometric functions need to be simplified. The integration of non-trigonometric functions is an important function. It is a common technique where the substitution rule with a trigonometric function is used then the resulting integral with a trigonometric identity is simplified.

Question 2: What is the use of an equilateral triangle to find sin 30 or 60 in geometry?

Answer: An equilateral triangle has all the sides and internal angles equal to each other i.e., 60 degrees. Thus, if the internal angle is 60 degrees then the external angle would be 180 - 60, that is 120 degrees.We are aware of the fact that 120 is one-third of 360 degrees. If we drop a base of an equilateral triangle (base being horizontal) then it forms a right angled triangle such that the new length of the base would be half of the original length. 

From trigonometry, cosine =‘adjacent/ hypotenuse’. 

Adjacent (base) is equal to half of the side. 

Thus base / hypotenuse would give us cosine(60)=0.5. 


Now,

We know that cos(60)²+ sin(60)² = 1. 

Thus, 

Sin(60) = √(0.5) = 0.707 rounded off.

Question 3: What is the value of sinθ.cosecθ?

Answer: The product of two inverse ratios is always 1. Thus, sin.cosec is equal to 1.

Question 4: What are the conditions of trigonometric ratios of positive angle?

Answer: The trigonometrical ratios of positive acute angle are always non- negative and 

  1. Sinθ and Cosθ can never be greater than 1.

  2. Cosecθ and Secθ can never be less than 1.

  3. Tanθ and Cotθ can have any value.  

Question 5: What is the opposite of Sin?

Answer: Cosec is the opposite of Sin.