Special Identities: verify tan(a/2) = (1-cosa)/(sina)

[zuuberbat]

a) Use the half-angle formulas for sine and cosine to verify the identity:
tan(a/2) = (1-cosa)/(sina)

b) Let x represent a real number. Find all solutions of the trigonometric equation:
tan(x/2) + sinx = 0

Explanations, steps would be helpful. Thank you!

 

[skeeter]

Re: Special Identities question...

a) Use the half-angle formulas for sine and cosine to verify the identity:
tan(a/2) = (1-cosa)/(sina)

\(\displaystyle \tan\left(\frac{a}{2}\right) = \frac{\sin\left(\frac{a}{2}\right)}{\cos\left(\frac{a}{2}\right)}\)

so ... use the half angle identities for sine and cosine as stated in the directions and see what happens.

b) Let x represent a real number. Find all solutions of the trigonometric equation:
tan(x/2) + sinx = 0

using the identity from part (a) ...

\(\displaystyle \frac{1-\cos{x}}{\sin{x}} + \sin{x} = 0\)

\(\displaystyle \frac{1-\cos{x} + \sin^2{x}}{\sin{x}} = 0\)

\(\displaystyle \frac{1-\cos{x} + (1 - \cos^2{x})}{\sin{x}} = 0\)

set the numerator = 0, solve the quadratic for \(\displaystyle \cos{x}\), then determine the value(s) for x.

 

[PAULK]

Re: Special Identities question...

a) Use the half-angle formulas for sine and cosine to verify the identity:
tan(a/2) = (1-cosa)/(sina)

\(\displaystyle \tan\left(\frac{a}{2}\right) = \frac{\sin\left(\frac{a}{2}\right)}{\cos\left(\frac{a}{2}\right)}\)

so ... use the half angle identities for sine and cosine as stated in the directions and see what happens.

Of course, you will write:

tan (a/2) = sqrt((1 - cos a)/2)/sqrt((1 + cos a)/2)

and you will proceed to cancel the 2's, then rationalize, then write sqrt( sin^2 a) = sin a.

But to do a REALLY good job, you will analyze how the signs are handled in the four quadrants. Then you will show you REALLY know your trigonometry.