Question #fe620

1 Answer
Apr 25, 2017

Answer:

#cos 2 theta = cos^2 theta - sin ^2 theta#

#tan 2 theta = (2 tan theta)/(1-tan^2 theta)#

Explanation:

You need to know what the angle #theta# is.

For example, if #theta = 30^@#
The ratio of a #30^@-60^@-90^@# triangle is #1: sqrt(3): 2#

So #sin 30^@ = 1/2; sin^2 30^@ = (1/2)^2 = 1/4#

#cos 30^@ = sqrt(3)/2; cos^2 30^@ = (sqrt(3)/2)^2 = 3/4#

#tan 30^@ = 1/sqrt(3); tan^2 30^@ = (1/sqrt(3))^2 = 1/3#

Use the trigonometric double angle identities:

#cos 2 theta = cos^2 theta - sin ^2 theta#

#cos 2*30^@ = 3/4 - 1/4 = 2/4 = 1/2#

#tan 2 theta = (2 tan theta)/(1-tan^2 theta)#

#tan 2*30^@ = (2 *1/sqrt(3))/(1-1/3) = (2/sqrt(3))/(2/3) #

#= 2/sqrt(3) * 3/2 = 3/sqrt(3) * sqrt(3)/sqrt(3) = (3 sqrt(3))/3 = sqrt(3)#