Sin 120


Sin 120 Degree

Trigonometry is one of the important topics in Mathematics and the concept of trigonometry is introduced by Greek Mathematician Hipparchus. The trigonometry basically deals with the angles and sides of a right-angle triangle. It is one of those parts of mathematics that helps in finding the missing sides and angles of a triangle with the help of trigonometry. The trigonometry angles are usually measured in radians or degrees. The most commonly used trigonometry angles are 0°, 30° 45°, 60°, 80°,120°,180°, etc. Trigonometry is widely used in various fields such as architecture, navigation system, sound waves detection etc. In this article, we will discuss how to find sine 120 degree exact values using unit circle and also we will learn how to find the value of sin 120  degrees other than a unit circle, solved examples based on sin 120 etc. 


What is the Sin 120 Degrees Value?

The Sin 120 Degrees value is \[\frac{\sqrt{3}}{2}\]

Hence, Sin 120 = \[\frac{\sqrt{3}}{2}\]


How to Derive Sin 120 Value?

Sin 120 value can be determined through the unit circle and the other trigonometry angles such as 60°, 180° etc. Let us examine the sin 120 value in the cartesian plane. As we know, the cartesian is divided into four quadrants. The sin 120 value comes under the second quadrant. All the values that come under the second quadrant take positive values. Hence, the sin 120 degrees value should be positive.

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On the basis of the above diagram of a unit circle, we can say that the value of sin 60 is equal to the value of sin 120 degrees.

It Implies that sin 60 = sin 120 degree =  \[\frac{\sqrt{3}}{2}\].

Hence, sin 120 degrees exact value is \[\frac{\sqrt{3}}{2}\].


What are the Methods to Derive the Value of Sin 120 Degrees Other than the Unit Circle?

There are two methods to derive the value of sin 120 degrees other than the unit circle. These are:


Method 1

Other than the unit circle,the value of sin 120 degree can be determined using other angles such as 60 degrees and 180 degrees which are derived from trigonometry tables.

As we know,

180°- 60° = 120°

We are also aware of the trigonometric identity sin(180°- a) = Sin a

Now,

Sin (180°- 120°) = Sin 120 degrees

Hence, sine 120 degrees = sin 60 degree

From the trigonometry table, we can see that the value of sin 60 degree is equals to \[\frac{\sqrt{3}}{2}\].

Hence, the value of sin 120 degree is \[\frac{\sqrt{3}}{2}\].


Method 2

Another method to derive the value of sine 120 degrees is by using cosine functions.

With the help of the trigonometry formula, sin (90 + a) = cos a ,we can determine sin 120 exact value.

As we know, sin (90° + 30°) = Sin 120 degrees.

Hence, sin 120 degree = cos 30°.

As we know the value of cos 30° is equal to \[\frac{\sqrt{3}}{2}\].

Hence, the sin 120 degrees exact value is \[\frac{\sqrt{3}}{2}\].


 Trigonometry Ratios Value Table

Angles in Degrees

0

30

45

60

90

Sin

0

\[\frac{1}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{\sqrt{3}}{2}\]

1

Cos

1

\[\frac{\sqrt{3}}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{1}{2}\]

0

Tan

0

\[\frac{1}{\sqrt{3}}\]

1

\[\sqrt{3}\]

Not defined

Cosec

Not defined

2

\[\sqrt{2}\]

\[\frac{2}{\sqrt{3}}\]

1

Sec

1

\[\frac{2}{\sqrt{3}}\]

\[\sqrt{2}\]

2

Not defined

Cot

Not defined

\[\sqrt{3}\]

1

\[\frac{1}{\sqrt{3}}\]

0


Solved Examples

1. Find the Value of Sin 120 - Cos 30

Solution:

The value of sin 120 = \[\frac{\sqrt{3}}{2}\]

The value of cos 30 =  \[\frac{\sqrt{3}}{2}\]

Hence, Sin 120 - Cos 30 =  \[\frac{\sqrt{3}}{2}\] -  \[\frac{\sqrt{3}}{2}\] = 0


2. Evaluate the Value of 3 Sin 30 + Tan 45

Solution:

The value of sine 120 = \[\frac{1}{2}\]

Value of tan 45  = 1

By substituting the values, we get

3 (\[\frac{1}{2}\]) + 1

= \[\frac{3}{2}\] + 1

= \[\frac{5}{2}\]


3. Find Sin 120°, Cos 120° and Tan 120°

Solution: 

Sin 120° = Sin(180°- 120°) =   \[\frac{\sqrt{3}}{2}\])

Cos 120° = -cos(180° - 120°) = - \[\frac{1}{2}\]

Tan 120° = Sin 120°/Cos 120° 

( \[\frac{\sqrt{3}}{2}\]) (\[\frac{1}{2}\]) = -\[\sqrt{3}\]


Quiz Time

1. The Sin Rule of a Triangle States that

  1. \[\frac{p}{Sin P}\] = \[\frac{q}{Sin Q}\] = \[\frac{r}{Sin R}\]

  2. \[\frac{P}{Sin p}\] = \[\frac{Q}{Sin q}\] = \[\frac{R}{Sin r}\]

  3. \[\frac{p}{Sin P}\] + \[\frac{q}{Sin Q}\] + \[\frac{r}{Sin R}\]

  4. \[\frac{2a}{Sin A}\] + \[\frac{2b}{Sin B}\] = \[\frac{2C}{Sin C}\]


2. Sin(A+ 45°) Sin(A-45°) is Equals to

  1. -\[\frac{1}{2}\]Cos(2A)

  2. -\[\frac{1}{2}\]Sin (2A)

  3. \[\frac{1}{2}\]Cos(2A)

  4. None of the above

FAQ (Frequently Asked Questions)

1. Explain Sine Function

The sine function is one of the most important functions in trigonometry other than cosine and tangent function The sine function is defined as the ratio of the length of the opposite side of the right-angle triangle to its hypotenuse side. For example, a triangle 

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ABC with an angle alpha, the sine function will be represented as :

Sin α- Opposite Side/ Hypotenuse side 

For the above triangle, the sin function will be defined as 

Sin α- BC/AB 

Hence, the sine formula for the above triangle is 

Sin α- a/h

2. Explain the Law of Sines and its Formula

Generally, the law of sine is defined as the ratio of the length of the sides of a triangle to the sine of the opposite angle. The law is sine is true for all three sides of a triangle irrespective of their sides and angles.

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a/Sin α = b/Sin β = c/Sin γ 

The unknown length of angle of a right-angle triangle can be determined through Sine rule. Sine rule, Sine law or Sine formula are the other names of law of sine.


Law of Sine Formula

a/Sin α = b/Sin β = c/Sin γ

a:b:c = Sin α = Sin β = Sin γ

a/b = Sin  α/Sin β

b/c =Sin β/Sin γ

It represents that if we divide a by the sine of ∠ α, it will be equal to the division of sine b by the Sine of ∠β and also equals to the division of sine c by Sine of ∠γ.