Half Angle Formulae

Introduction

Trigonometric identities are relationships between quantities (angles) that are likely for all values of the involved variables. Geometrically speaking, these identities are of one or more angles or perhaps even half or quarter of an angle. 

There are several trigonometric identities. First and foremost is the Pythagorean theorem, which involves a circle of unit one.

x2+y2=1, whose trigonometric counterpart happens to be sin2+cos2=1.

From the Pythagorean theorem for the unit circle, it can be thus said that:

sin=√1-cos2

cos=√1-sin2, where the symbol + or – depends on the quadrant of the circle the angle theta is in. 

There are addition and subtraction identities, followed by multiple angle formulae and power of angle formulae.

Simple calculations

Suppose you are asked to calculate the value of 7.5 degrees using half-angle formulae for sine, cosine, and tangent. You know that 7.5 is the half of 15 degrees and that 15 degrees fall under the half of 30 degrees. Now, you already know the value of 30 degrees. 

The half angle formulae equations are as follows:

sin2(/2)=(1-cos)/2

cos2(/2)=(1+cos)/2

tan(/2)=(1-cos )sin =sin (1+cos )

The tangent’s half-angle formula does not require a plus or a minus sign, unlike those for the sine or the cosine.

The function for sine can be first calculated as 30/2 and then 15/2. The same goes for cosine and tangent. 

The Area of a Right-Angled Triangle using Trigonometry

As we already know, 12baseheight gives the area of a right triangle.

However, there are other ways to calculate the area of a right triangle if you are aware of the angle formulae.

Let us consider a triangle ABC in which: 

AC = b (the larger hypotenuse)

AB = c (the smaller hypotenuse)

AD = h (the height of the right-angled triangle)

BC = a (the base of the triangle).

Using the right triangle formula for angle C,

sin C=ADAC

sin C=hb

h=b sin C

Therefore, area=ab sin C2

This is the derivation for the right triangle formula.

Trigonometric Identities

Half-angle formulae are derived from multiple rather than double angle formulae. The double angles in trigonometry can be obtained using the sum and difference formulae. However, to obtain the half-angle formulae, a different set of identities must be used.

sin =2 sin (2) cos 2

cos =cos2(2)-sin2(2)

=1-2 sin2(2)

=2 cos2(2)-1

tan =(2tan(/2))/1-tan2(/2)

Half Angle Formula using a Scalene triangle

We have a triangle ABC in which, 

AB = c

BC = a

AC = b

Half the perimeter is calculated by S=(a+b+c)2.

The half angle formulae for each of the angles are given as follows:

sin A2=((s-b)(s-c)bc)12

sin C2=((s-a)(s-b)ab)12

cos A2=(s(s-a)bc)12

cos B2=(s(s-b)ca)12

cos C2=(s(s-c)ab)12

tan A2=((s-b)(s-c)s(s-a))12

tan B2=((s-c)(s-a)s(s-b))12

tan C2=((s-a)(s-b)s(s-c))12

These equations, as mentioned above, are the angle formulae for the three angles A, B, and C, the ones opposite the sides BC, AC, and AB.

Conclusion

Sometimes we replace sine and cosine functions with tangent half-angle formulae as a completely different variable designated as ‘t.’ The tangential function of the bisection of an angle is the projection of a unit circle on a straight line. This is evident once you consider the Pythagoras theorem. Additionally, this replacement is used in integral calculus. They are also a part of hyperbolic functions. This is known as the Weierstrass Substitution, in which sine and cosine functions of x (as integrals) are converted into functions of t with t being equivalent to t = tan (x/2). The value of x ranges from minus pi to plus pi in half a circle. 

The integral substitution goes thus:

sin (x2)=t(1+t2)12 

cos (x2)=1(1+t2)12 

Therefore, 

sin x=2t(1+t2) and cos x=(1-t2)(1+t2)

dx=(2(1+t2)) dt

This substitution aims to change the limits of the integration from infinity to so that it becomes much easier to rationalize and integrate.