Properties of half Angle formula

Maths is a vast field that helps us learn more about the world that we are a part of. Within Maths, there can be many sub-fields and here we shall be focusing on one such significant area of Maths, that is, trigonometry. One can say that that area of Maths that focuses on topics pertaining to the dimensions, angles and measures can be called trigonometry. Here, we will be specifically focusing on the properties of the half-angle formula. 

Angle

Ever thought about what exactly is an angle? Well, when there are two lines at some inclination with one another then it can be said that an angle has formed. The angle’s inclination can be called its measurement. At times, a dimension gets formed due to many angles coming together. 

One can learn about various ways for finding the angle’s measurement. The measurement for trigonometric 180 degrees, 60 degrees, 0 degrees etcetera for sin, tan, cos, sec, cot and cosec are already there. Now, in the same manner, there is something known as the half-angle formula in Maths. 

Half Angle Formulae

So as to figure out the measurement for angles that differ from the renowned values such as zero degrees, ninety degrees etcetera, half angles can be important. Double angle formulas can be used for deriving the half angles. Thus, since half angles can be derived using the double angle formulas, these are given below as: – 

1. Cos X/2 = +/- ((1 + Cos X)/2)¹⁄² 
2. Tan X/2 = (1 – Cos X) SinX
3. Sin X/2 = +/- ((1 – Cos X)/2)¹⁄²

Apart from these, the other significant identities regarding the double angle formula that can be used for deriving the half-angle formula are as follows: – 

   Cos2 X = Cos (X + X)

Cos2X = (CosX)(CosX) – (SinX)(SinX)

Cos2X = Cos² X – Sin² X

Cos2X = Cos² X – (1 – Cos²X)

Cos2X = 2Cos²X – 1 ……………. This can be the equation 1 

Now, putting Cos²X = 1 – Sin²X

Cos2X = 2(1 – Sin²X) – 1

Cos2X = 2 – 2Sin²X – 1

Cos2X = 1 – 2Sin²X……………. This can be the equation 2 

Thus, with the help of the above equations the derivation for the half-angle formula for cos, half-angle formula for sin and half-angle formula for tan can be conveniently done. 

After completing the derivation process, the half-angle formula for cos can be such that: – 

Cos(y/2) = +/- √ (1+ Cosy)/2

After completing the derivation process, the half-angle formula for sin can be such that: – 

Sin(y/2) = +/- √ (1+ Cosy)/2

After completing the derivation process, the half-angle formula for tan can be such that: – 

Tan(y/2) = +/- √ (1- Cosy)/(Siny)

Important Points

The above section helped in learning more about the derivation process of the half-angle formula along with the three different half-angle formulas for sin, cos and tan. Now here, we will discuss some significant properties regarding the half-angle formula. These include – 

1. Double angle formulas are used for the derivation process of half-angle formulas. 
2. The angle difference and the angle sum formulas in trigonometry are used for the derivation of the double angle formula.
3. Half angle formulas can be very helpful for finding the precise value regarding the various trigonometric ratios such as fifteen degrees, twenty-two point five degrees etcetera. 

Thus, it can be undoubtedly said that half-angle formulas have a very important role to play in trigonometry. 

Conclusion

As observed from the above sections, half-angle formulas are a significant part of trigonometry which is again a significant area of Maths. In the above sections, various topics have been covered. These topics help in understanding the role of the half-angle formula in trigonometry, the derivation process for the half-angle formula in trigonometry and the process of using the half-angle formula in trigonometry. It can be observed that double angle formulas can be used for deriving the half-angle as well. Thus, last it can be concluded that the above sections helped in learning about the half-angle formula of trigonometry in a comprehensive and clear manner.