eguruchela

Trigonometry identities Power Reduction Calculator

The Trigonometry Identities Power Reduction Calculator computes sin2u, cos2u and tan2u for given angle


 

 

using following formulas:

sin2u = 1/2 - (1/2)cos(2u))

cos2u = 1/2 + (1/2)cos(2u))

tan2u = (1 - cos(2u)) / (1 + cos(2u))

sin3u = (3/4)sinu - (1/4)sin(3u)

cos3u= (3/4)cosu + (1/4)cos(3u) and so on ..

The power reduction formulas allows to transform sin2(u) and cos2(u) into expressions that contains the first power of cosine of double argument.

These functions are in the same way as double-angle and half-angle functions. The power reduction formulas can be derived through the use of double-angle, half-angle formulas and Pythagorean Identity.

The use of a power reduction formula expresses the quantity without the exponent.

Proof the power reduction formula for sin and cosin

Proof for sin :

cos(2u) = cos2u - sin2u ............(1)

we will use the Pythagorean Identities as sin2u + cos2u = 1

so cos2u = 1 - sin2u, we can substitute the value of cos2u in equation (1) and we will get as :

cos2u = (1 - sin2u) - sin2u

cos2u = (1 - 2sin2u)

Now sustract 1 from the both sides

cos2u - 1 = (1 - 2sin2u) - 1

2sin2u = 1 - cos2u

sin2u = (1 - cos2u)/2

sin2u = 1/2(1 - cos2u)

 

Proof for cos :

cos(2u) = cos2u - sin2u ............(1)

We will use the Pythagorean Identities as sin2u + cos2u = 1

so sin2u = 1 -cos2u , we can substitute the value of sin2u in equation (1) and we will get as :

cos2u = cos2u - (1 -cos2u)

cos2u = 2cos2u - 1

Now add 1 from the both sides

cos2u + 1 = 2cos2u - 1 + 1

cos2u + 1 = 2cos2u

2cos2u = cos2u + 1

cos2u = 1/2(cos2u + 1)