Learn the Value of Sin 30 Degrees
Trigonometry is not only important to score high marks in mathematics but also in day-to-day life. Trigonometry starts with the most important functions of ratio and reciprocal. Trigonometric ratios are calculated only for right angled triangles
Sine, cosine and tangent are the three main pillars on which the whole concept of trigonometry rests. Sin is one of those important trigonometric ratios. The value of sin 30 degrees is half (½). Just to recapitulate the sides of a triangle, let's go through the following definitions once again since this will help the students to understand the sides and their ratios in relevance to trigonometry.
Trigonometric Ratios
Trigonometric ratios are used to calculate the unknown sides or angles of a triangle that can not be calculated from the simple properties of triangles. However this is only applicable for right angled triangles where the ratios of sides are expressed in the form of six trigonometric ratios. They are 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.
Reciprocals:
The reciprocal or inverse of Sin is Cosec. That is,
If Sinϴ = Perpendicular / Hypotenuse then Cosec ϴ = Hypotenuse /Perpendicular
The reciprocal or inverse of Cos is Sec. That is,
If Cosϴ = Base / Hypotenuse then Sec ϴ = Hypotenuse/ Base
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
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:
Two angles and one side of a triangle given.
Two sides and one included angle 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/AB = a/2a = 12
Thus, the value of Sin 30 degrees 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°.
Vedantu has arranged the chapter of trigonometry with utmost care with lots of examples and derivations done by subject teachers in an easy understandable way. They have given special focus on each function separately like here for sin30 degrees.
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 .
FAQs on Sin 30 Degrees
1. What are the trigonometric identities?
Trigonometric identities are basically equations which involve trigonometric functions and are true for each and every value of the variables involved in the function present on both sides of the equation. They are applicable only for right angled triangles. The most common trigonometric identities are derived from the Pythagoras Theorem. They help to simplify complex trigonometric functions of one or more angles. The fundamental trigonometric identities are based on the six most important trigonometric ratios. The trigonometric identities are extremely useful in solving any kind of trigonometric problems in a simple way.
2. What is the use of an equilateral triangle to find sin 30 or 60 in geometry?
An equilateral triangle means a triangle with three equal sides and three internal angles equal to each other. This also implies the internal three angles are of 60 degrees each since we know the sum of the internal angles of a triangle is always 180 degrees. If the internal angle is 60 degrees then it means that the external angle would be 120 degrees since a total angle of a line is 180 degrees. Now we also know that 120 degrees means one-third of 360 degrees. Now if any one base of the equilateral triangle is dropped we will get a right angled triangle with the length of the base now being just half of the actual length.
We know that cosine is equal to the ratio of adjacent sides and the hypotenuse.
Since the adjacent base is now the half of the side,
We get (60)=0.5. Since base/hypotenuse=1/2
We also know from trigonometry that cos(60)2+ sin(60)2 = 1.
Therefore, after substitution and calculation we get
Sin(60) = √(0.5) = 0.707 approx.
3. What are the more important trigonometric identities?
In fact all the trigonometric identities are important but few are more important than the others which forms the base of all the identities and functions. They are listed as follows:
Relations for tangent and secant in terms of sine, cotangent and cosecant in terms of cosine.
Expression of sine and cosine in terms of Pythagorean formulae.
Periods of sine, cosine, secant, cosecant which is 2𝝿 and periods of tangent and cotangent which is 𝝿.
Identities of negative angles: Except for cosine and secant all are odd functions.
Formulas for sine and cosine double angles. Three of them to be learnt.
Ptolemy’s identities.
The Formulae of sum and difference for sine and cosine
4. What are the main trigonometric functions?
The three main pillars of trigonometric functions are sine, cosine and tangent. They are expressed as:
Sin θ = Opposite side/hypotenuse
Cos θ = Adjacent side/hypotenuse
Tan θ = Opposite side/ Adjacent side.
All the identities and trigonometric problems revolve around these three main functions.
5. What is the importance of studying trigonometric functions?
Trigonometry is one of the most important topics in the field of measurements. It is used to find the unknown sides and angles of a right angled triangle from the relationships between angles, lengths, heights. The measures which can not be found by simple principles of triangles can be found by different trigonometric ratios. The importance lies in the real world also especially for finding periodic phenomena like waves, sound and light. Also used in celestial mechanics and solid mechanics. They are especially important for periodic phenomena with the use of Fourier analysis. In geography it is used to measure long distances between different landmarks. It is also used to find the distance of nearby stars and satellites. It is also important to calculate the force that would be created on bridges and girders. Interestingly it is used in criminology to find the angle in which a bullet is shot or to find the reason behind a car collision and many other things.
