Sin 30 Degrees: Value, Derivation & Trigonometry Table

Sin 30 degrees has a value of ½ or 0.5. In radians, Sin 30^o is represented as Sin \(\frac{\pi}{6}\). Sine or Sin is one of the most important trigonometric ratios whose values range between -1 to +1. The other important trigonometric ratios are Cos, Tan, Sec, Cosec, Cot. In this article, we will learn about the value of Sin 30^o and its derivation.

Read Here: Applications of Trigonometry

Key Takeaways: Sin 30 degrees, Radians, Trigonometric Ratios, Trigonometric Table, Geometric Derivation.

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Sin 30 Degrees Value 

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In a right-angled triangle, the sin of the angle is the ratio of the side opposite the angle to the hypotenuse (longest side of triangle).

\(Sin \theta = \frac{Opposite}{Hypotenuse}\)

The angle sin 30° is between 0° and 90° and lies in the First Quadrant and the value for sin 30 degree angle is 0.5.

Sin 30 Degrees

Read Here: Introduction to Trigonometry

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Geometric Derivation of Sin 30 Degree 

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Consider an equilateral triangle ABC

We know that, All angles in an equilateral triangle are 60 degrees.

Thus , ∠A = ∠B = ∠C = 60^o

Now draw a perpendicular line AD from A to BC as shown in the figure below.

Now in Triangle ΔABD = ΔACD

Therefore,       BD = DC

Also ΔBAD = ΔCAD (By Corresponding Parts of Congruent Triangles)

Lets assume the length of side of the equilateral triangle  AB = BC = AC = 2a.

Then, BD = ½ BC = ½ x 2a = a

 In Right Triangle ADB,

AD² + BD² = AB² [By Pythagoras theorem]

 AD² = = AB²- BD²

(2a)² – a² = (3a)²

AD² = 3a²

AD = a√3

Now, In right Triangle ADB,

Sin 30^o = BD/AB = a/2a = ½ or 0.5.

Similarly, we can represent Sin 30^o  by using trigonometric identities as follows.

• Sin (180^o – 30^o) = Sin 50^o
• -Sin (180^o + 30^o) = -Sin 210^o
• Cos (90^o - 30^o) = Cos 60^o
• -Cos (90^o + 30^o) = -Cos 120^o

Check More: Trigonometry Important Formulae

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Trigonometric Values Table 

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The important values of trigonometric ratios are given in the table below.

Also Read:

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Things to Remember 

• The value of Sin 30 degrees is ½ or 0.5.
• Sin 30 degrees is also written as Sin \(\frac{\pi}{6}\) or sin (0.523598..)
• The decimal value of sin 30 degree and Cos 60 degree is same i.e. 0.50
• Sin 30° is lying in the first quadrant so its final value will be positive.
• Sinθ reciprocal is Cosecθ. We also say that Sinθ = Perpendicular / Hypotenuse, then Cosecθ = Hypotenuse / Perpendicular.
• For the conversion of degree to radian we use, θ in radians = θ in degree* (Pi/180).

Read More: Cosine Rule Important Notes

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