What is the Half-Angle Identities?
1 Answer
The half-angle identities are defined as follows:
\mathbf(sin(x/2) = pmsqrt((1-cosx)/2))
(+) for quadrants I and II
(-) for quadrants III and IV
\mathbf(cos(x/2) = pmsqrt((1+cosx)/2))
(+) for quadrants I and IV
(-) for quadrants II and III
\mathbf(tan(x/2) = pmsqrt((1-cosx)/(1+cosx)))
(+) for quadrants I and III
(-) for quadrants II and IV
We can derive them from the following identities:
sin^2(x/2) = (1-cos(x))/2
color(blue)(sin(x/2) = pmsqrt((1-cos(x))/2))
Knowing how
cos^2(x/2) = (1+cos(x))/2
color(blue)(cos(x/2) = pmsqrt((1+cos(x))/2))
Knowing how
color(blue)(tan(x/2) = pmsqrt((1-cos(x))/(1+cos(x))))
We can see that if we take the conditions for positive and negative values from