Sin 120

Trigonometry is one of the important topics in Mathematics and the concept of trigonometry is introduced by Greek Mathematician Hipparchus. 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 systems, sound waves detection, etc. In this article, we will discuss how to find sine 120-degree exact values using a unit circle and also we will learn how to find the value of sin 120  degrees other than a unit circle, solve examples based on sin 120, etc. 

How to Derive Sin^o 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.

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

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

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

What are the Methods to derive the Value of sin^o 120 degrees Other than the Unit Circle?

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

Method 1

Other than the unit circle, the value of sin^o 120 degrees 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^o 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 degrees is \[\frac{\sqrt{3}}{2}\].

Method 2

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

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

As we know, sin^o (90° + 30°) = sin^o 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

Solved Examples

1. Find the value of sin 120^o - cos 30^o

Solution: The value of sin^o 120 =\[\frac{\sqrt{3}}{2}\]

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

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

2. Evaluate the value of 3 sin 30^o + tan 45^o

Solution:The value of sine 120^o = \[\frac{1}{2}\]

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

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

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

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

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

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

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

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

4. None of the above

Conclusion

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