# ✅ Sin 30 Formula ⭐️⭐️⭐️⭐️⭐️

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Sin 30 Formula: Sin 30 degrees is equal to cos 60 degrees. The value of sin 30 degrees is 1/2.

What is the formula for sin 30?

In an equilateral triangle, the measure of each triangle is 60° and all the sides are equal.

Let’s assume all the side to be of 2 cm. After bisecting the angle, it would become 30°and length of the side will be ½ (the side opposite to the angle 60 degrees which equal to 1), this way we get sine 30° as ½.

In a right angle triangle, one of the angles is 90°. To find the value of various trigonometric functions for different angles, we refer to the trigonometry table.

Mục Lục

## What is the Value of Sin 30°?

The value of sin 30 degrees can be found with the help of the above trigonometric table.

## Solved Examples

Example 1: Find the value of sin 30° + 2 cos 60°.

Solution:
sin 30° + 2 cos 60°
= sin 30° + 2 cos (90° – 30°)
= sin 30° + 2 sin 30°
= 3 sin 30°
= 3(½)
= 3/2

Example 2: Simplify: 2 sin 30°/(1 – sin230°)

Solution:
2 sin 30°/(1 – sin230°)
= 2 (½)/[1 – (½)2] = 1/[1 – ¼] = 1/(¾)
= 4/3

Example 3: Find the value of 5 sin 30°/7 cos 60°.

Solution:
5 sin 30°/7 cos 60°
= 5 sin 30°/ 7 cos (90° – 30°)
= (5/7) (sin 30°/sin 30°)
= (5/7)(1)
= 5/7

## Sin 30 degrees

The value of sin 30 degrees is 0.5. Sin 30 is also written as sin π/6, in radians. The trigonometric function also called as an angle function relates the angles of a triangle to the length of its sides. Trigonometric functions are important, in the study of periodic phenomena like sound and light waves, average temperature variations and the position and velocity of harmonic oscillators and many other applications. The most familiar three trigonometric ratios are sine function, cosine function and tangent function.

For angles less than a right angle, trigonometric functions are commonly defined as the ratio of two sides of a right triangle. The angles are calculated with respect to sin, cos and tan functions. Usually, the degrees are considered as 0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°. Here, we will discuss the value for sin 30 degrees and how to derive the sin 30 value using other degrees or radians.

## Sine 30 Degrees Value

The exact value of sin 30 degrees is ½. To define the sine function of an angle, start with a right-angled triangle ABC with the angle of interest and the sides of a triangle. The three sides of the triangle are given as follows:

• The opposite side is the side opposite to the angle of interest.
• The hypotenuse side is the side opposite the right angle and it is always the longest side of a right triangle
• The adjacent side is the side adjacent to the angle of interest other than the right angle

The sine function of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side and the formula is given by:

Sine Law: The sine law states that the sides of a triangle are proportional to the sine of the opposite angles.

The sine rule is used in the following cases :

Case 1: Given two angles and one side (AAS and ASA)

Case 2: Given two sides and non included angle (SSA)

The other important sine values with respect to angle in a right-angled triangle are:

Sin 0 = 0

Sin 45 = 1/√2

Sin 60 = √3/2

Sin 90 = 1

Fact: The values sin 30 and cos 60 are equal.

Sin 30 = Cos 60 = ½

And

Cosec 30 = 1/Sin 30

Cosec 30 = 1/(½)

Cosec 30 = 2

## Derivation to Find the Sin 30 value (Geometrically)

Let us now calculate the sin 30 value. Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore

Draw the perpendicular line AD from A to the side BC (From figure)

Now observe that the triangle ABD is a right triangle, right-angled at D with  ∠BAD = 30° and ∠ABD = 60°.

As you know, for finding the trigonometric ratios, we need to know the lengths of the sides of the triangle. So, let us suppose that AB=2a

To find the sin 30-degree value, let’s use sin 30-degree formula and it is written as:

Sin 30° = opposite side/hypotenuse side

We know that, Sin 30° = BD/AB = a/2a = 1 / 2

Therefore, Sin 30 degree equals to the fractional value of 1/ 2.

Sin 30° = 1 / 2

Therefore, sin 30 value is 1/2

In the same way, we can derive other values of sin degrees like 0°, 30°, 45°, 60°, 90°,180°, 270° and 360°. Below is the trigonometry table, which defines all the values of sine along with other trigonometric ratios.

### Why Sin 30 is equal to Sin 150

The value of sin 30 degrees and sin 150 degrees are equal.

Sin 30 = sin 150 = ½

Both are equal because the reference angle for 150 is equal to 30 for the triangle formed in the unit circle. The reference angle is formed when the perpendicular is dropped from the unit circle to the x-axis, which forms a right triangle.

Since, the angle 150 degrees lies on the IInd quadrant, therefore the value of sin 150 is positive.The internal angle of triangle is 180 – 150=30, which is the reference angle.

The value of sine in other two quadrants, i.e. 3rd and 4th are negative.

In the same way,

Sin 0 = sin 180

### Trigonometry Table

 Trigonometry Ratio Table Angles (In Degrees) 0 30 45 60 90 180 270 360 Angles (In Radians) 0 π/6 π/4 π/3 π/2 π 3π/2 2π sin 0 1/2 1/√2 √3/2 1 0 −1 0 cos 1 √3/2 1/√2 1/2 0 −1 0 1 tan 0 1/√3 1 √3 Not Defined 0 Not Defined 0 cot Not Defined √3 1 1/√3 0 Not Defined 0 Not Defined cosec Not Defined 2 √2 2/√3 1 Not Defined −1 Not Defined sec 1 2/√3 √2 2 Not Defined −1 Not Defined 1

## Solved Examples

Question 1: In triangle ABC, right-angled at B, AB = 5 cm and angle ACB = 30°. Determine the length of the side AC.

Solution:

To find the length of the side AC, we consider the sine function, and the formula is given by

Sin 30°= Opposite Side / Hypotenuse side

Sin 30°= AB / AC

Substitute the sin 30 value and AB value,

Therefore, the length of the hypotenuse side, AC = 10 cm.

Question 2: If a right-angled triangle has a side opposite to an angle A, of 6cm and hypotenuse of 12cm. Then find the value of angle.

Solution: Given, Side opposite to angle A = 6cm

Hypotenuse = 12cm

By sin formula we know that;

Sin A = Opposite side to angle A/Hypotenuse

Sin A = 6/12 = ½

We know, Sin 30 = ½

So if we compare,

Sin A = Sin 30

A = 30

Hence, the required angle is 30 degrees.

Question 3: If a right-angled triangle is having adjacent side equal to 10 cm and the measure of angle is 45 degrees. Then find the value hypotenuse of the triangle.

Solution: Given adjacent side = 10cm

We know,

Tan 45 = Opposite side/Adjacent side

Tan 45 = opposite side/10

Since, Tan 45 = 1

Therefore,

1 = Opposite side/10

Opposite side = 10 cm

Now, by sin formula, we know,

Sin A = Opposite side/Hypotenuse

So,

Sin 45 = 10/Hypotenuse

Hypotenuse = 10/sin 45

Hypotenuse = 10/(1/√2)

Hypotenuse = 10√2

## Frequently Asked Questions – FAQs

### What is the exact value of sine 30 degrees?

The exact value of sin 30 degrees is 0.5.

### What is the value of sine 30 in the form of fraction?

The value of sin 30 in the form fraction is ½

### What is sine 30 degrees in radian?

Sine 30 degrees in radian is given by sin π/6.

### What is the formula for sine function?

Sine of an angle, in a right-angled triangle, is equal to the ratio of opposite side of the angle and hypotenuse.
Sine A = Opposite Side/Hypotenuse

### What is the value of cos 30 and tan 30?

The value of cos 30 degrees is √3/2 and the value of tan 30 degrees is 1/√3.

### How to find the value of tan 30 with respect to sin and cos?

Tangent of an angle is equal to the ratio of sine and cosine of the same angle. Therefore,
Tan 30 = sin 30/cos 30
We know that, sin 30 = ½ and cos 30 = √3/2
Therefore, tan 30 = (½)/√3/2
Tan 30 = 1/√3

## 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:

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.

### 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:

• 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,

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 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.

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 .

## What is the Value of Sin 30 Degrees?

The value of sin 30 degrees in decimal is 0.5. Sin 30 degrees can also be expressed using the equivalent of the given angle (30 degrees) in radians (0.52359 . . .).

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 30 degrees = 30° × (π/180°) rad = π/6 or 0.5235 . . .
∴ sin 30° = sin(0.5235) = 1/2 or 0.5

Explanation:

For sin 30 degrees, the angle 30° lies between 0° and 90° (First Quadrant). Since sine function is positive in the first quadrant, thus sin 30° value = 1/2 or 0.5
Since the sine function is a periodic function, we can represent sin 30° as, sin 30 degrees = sin(30° + n × 360°), n ∈ Z.
⇒ sin 30° = sin 390° = sin 750°, and so on.
Note: Since, sine is an odd function, the value of sin(-30°) = -sin(30°).

## Methods to Find Value of Sin 30 Degrees

The sine function is positive in the 1st quadrant. The value of sin 30° is given as 0.5. We can find the value of sin 30 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

## Sin 30 Degrees Using Unit Circle

To find the value of sin 30 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form a 30° angle with the positive x-axis.
• The sin of 30 degrees equals the y-coordinate(0.5) of the point of intersection (0.866, 0.5) of unit circle and r.

Hence the value of sin 30° = y = 0.5

## Sin 30° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin 30 degrees as:

• ± √(1-cos²(30°))
• ± tan 30°/√(1 + tan²(30°))
• ± 1/√(1 + cot²(30°))
• ± √(sec²(30°) – 1)/sec 30°
• 1/cosec 30°

Note: Since 30° lies in the 1st Quadrant, the final value of sin 30° will be positive.

We can use trigonometric identities to represent sin 30° as,

• sin(180° – 30°) = sin 150°
• -sin(180° + 30°) = -sin 210°
• cos(90° – 30°) = cos 60°
• -cos(90° + 30°) = -cos 120°

## Examples Using Sin 30 Degrees

Example 1: Find the value of 5 sin(30°)/7 cos(60°). Solution: Using trigonometric identities, we know, sin(30°) = cos(90° – 30°) = cos 60°.
⇒ sin(30°) = cos(60°)
⇒ Value of 5 sin(30°)/7 cos(60°) = 5/7

Example 2: Simplify: 2 (sin 30°/sin 390°)

Solution:

We know sin 30° = sin 390°
⇒ 2 sin 30°/sin 390° = 2(sin 30°/sin 30°)
= 2(1) = 2

Example 3: Using the value of sin 30°, solve: (1-cos²(30°)).

Solution: We know, (1-cos²(30°)) = (sin²(30°)) = 0.25
⇒ (1-cos²(30°)) = 0.25

## FAQs on Sin 30 Degrees

### What is Sin 30 Degrees?

Sin 30 degrees is the value of sine trigonometric function for an angle equal to 30 degrees. The value of sin 30° is 1/2 or 0.5.

### How to Find the Value of Sin 30 Degrees?

The value of sin 30 degrees can be calculated by constructing an angle of 30° with the x-axis, and then finding the coordinates of the corresponding point (0.866, 0.5) on the unit circle. The value of sin 30° is equal to the y-coordinate (0.5). ∴ sin 30° = 0.5.

### How to Find Sin 30° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin 30° can be given in terms of other trigonometric functions as:

• ± √(1-cos²(30°))
• ± tan 30°/√(1 + tan²(30°))
• ± 1/√(1 + cot²(30°))
• ± √(sec²(30°) – 1)/sec 30°
• 1/cosec 30°

☛ Also check: trigonometry table

### What is the Value of Sin 30° in Terms of Cosec 30°?

Since the cosecant function is the reciprocal of the sine function, we can write sin 30° as 1/cosec(30°). The value of cosec 30° is equal to 2.

### What is the Value of Sin 30 Degrees in Terms of Cot 30°?

We can represent the sine function in terms of the cotangent function using trig identities, sin 30° can be written as 1/√(1 + cot²(30°)). Here, the value of cot 30° is equal to 1.73205.

## Solved Examples on Sin 30 degree

Example 1: In a right-angled triangle, the side opposite the angle of 30° is 7m. Find the length of the Hypotenuse.

Solution:

Given: Perpendicular = 7m

Sin 30 = 1/2

P/H = 1/2

7/H = 1/2

H = 7 × 2

H=14m

Example 2: In a right triangle hypotenuse is 20cm, and one side is 10√3cm, find the angles of the triangle.

Solution:

Given: H=20, and B = 10√3

Finding third side using pythagoras theorem.

P2 + B2 = H2

P2 + (10√3)2 = 202

p2 + 300 = 400

P2 = 100

P = 10

The third side is 10cm. The ratio of the third side and the hypotenuse is 1/2 (10/20) So there must be an angle of 30° in triangle Since the triangle is a right angle, so the third angle is

90° – 30° = 60°

The Angles of a triangle are 30°, 60°, 90°.

Example 3: In a right-angled triangle adjacent side is 12 cm and the measure of an angle is 60°. Now find the value of the hypotenuse of the triangle.

Solution:

Given adjacent side = 12 cm

tan 60 = Opposite side/Adjacent side = opposite side/12         (tan 60 = √3)

Now,

1 = Opposite side/12

Opposite side = 12√3 cm

Now,

sin A = Opposite side/Hypotenuse

sin 60 = 10/Hypotenuse

Hypotenuse = 10/sin 60

Hypotenuse = 10/(√3/2)

Hypotenuse = 20√3/3

## FAQs on sin 30 degree

Question 1: Write the value of sine 30°.

The value of sin 30 degrees is 1/2 or 0.5.

Question 2: Write the value of 30 degrees in radian.

The value of 30 degrees in radian is π/6.

Question 3: Write the formula for sin function

In a right-angle triangle sin of an angle, is equal to the ratio of the opposite side of the angle and hypotenuse.

i.e  sin A = Opposite Side/Hypotenuse

Question 4: Write the value of cos 30° and tan 30°.

The value of cos 30° is √3/2 and the value of tan 30° is 1/√3.

Q. In the triangle XYZ, right-angled at Y, Sise XY is equal to 10 centimetres, and the angle XZY is equal to 30 degrees. Determine how long each of the sides XZ is.

A.

Sin 30°= Perpendicular Hypotenuse

Sin 30°= XY / XZ

Putting Sin 30 value in

½ = XY/ XZ

½ = 10/ XZ

XZ = 20cm

Therefore, XZ = 20 cm.

## Sample Questions

Ques: In a Right angle triangle ABC, Side AB = 24cm & side BC = 7 cm, Find
1) Sin A & Cos A
2) Cos C & Sin C

Ans: In triangle ABC,

Given: side AB = 24cm & BC = 7 cm

By using Pythagoras theorem,

AC2 = AB2+ BC2

AC2 = 242+ 72

AC2 = 576 + 49 = 625

AC = 25

Now for,

1). Sin A = BC/AC = 7/25

Cos A = AB/AC = 24/25.

2). Cos C =BC/AC = 7/25

Sin C = AB/AC =24/25.

Ques: In a Right angle triangle ABC, Side AB = 30cm & side BC = 10 cm, Find Sin A & Cos A

Ans: In triangle ABC,

Given: side AB = 30 cm & BC = 10 cm

By using Pythagoras theorem,

AC2 = AB2+ BC2

AC2 = 302+ 102

AC2 = 900 + 100 = 1000

AC = 31.62

Now for,

Sin A = BC/AC = 10/31.62 = 0.316

Cos A = AB/AC = 30/31.62 = 0.94

Ques: Find the value of Sin230 + tan245+ cot2 60

Ans: From the trigonometric table,

We know that,

Sin 30 = ½

Tan 45 = 1

Ques: Find the value of Sin230 + Sin245+ Sin2 60

Ques: Find tan A – cot C in figure?

Ans: In right angle triangle, ABC

By using Pythagoras theorem, we get

BC2 = AC2 – AB2

QR2 = 132 – 122

= (13-12) (13+12)

= 1* 25 = 25

BC2 = 25

BC = 5

Now,

Tan A = BC/AB = 5/12

Cot C = BC/AB = 5/12

So, tan A – cot C = 5/12 – 5/12 = 0

Ques: What is the value of sin(-30)?

Ans: We already know that the value of sin 30 = 0.5

sin (-30) = – sin (30)

therefore, sin (-30) = – 0.5

Ques: What is the value of sin 30 + cos 30?

Ques: What is the value of sin 30 + tan 30?

Ques: What is the value of sin 30 + Cos 60?

Ans: We already know from the trigonometric table,

Sin 30 = ½

Cos 60 = ½

Therefore,

½ + ½  = ¼

= 0.25

Ques: In right angled triangle ABC, Angle ACB = 30° and AB = 5 cm. Find the lengths of AC and BC sides?

Ans: Given: ∠ACB = 30° and side AB = 5 cm.

We know that,

Tan θ = Perpendicular/ Base

So, Tan C = AB/ BC or tan 30° = 5/ BC

1/ √3 = 5cm /BC (by using table, tan30° = 1/√3)

And also, BC = 5√3 cm.

Now, for third side by using trigonometric identities we get

Sin C = AB/ AC or Sin 30° = 5cm / AC

We know that Sin 30° = ½.

So, ½ = 5 cm /AC

AC = 10 cm.

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