How do you find the cos of theta if #abs(sintheta)=1#?

1 Answer
May 25, 2016

Answer:

#cos theta = 0#

Explanation:

No matter what the value of #theta#, the following holds:

#cos^2 theta + sin^2 theta = 1#

Considering #sin theta# as a Real valued function of Real numbers:

#abs(sin theta) = 1# implies #sin theta = +-1#, which implies #sin^2 theta = 1#

So:

#cos^2 theta = 1 - sin^2 theta = 1-1 = 0#

So:

#cos theta = 0#

#color(white)()#
Complex footnote

The same does not hold if #theta# can take Complex values.

For any value of #z# we can define:

#cos z = (e^(iz)+e^(-iz))/2#

#sin z = (e^(iz)-e^(-iz))/(2i)#

For example:

If #theta = ln(1+sqrt(2))i# then:

#sin(theta) = i#, so #abs(sin(i)) = 1#

#cos(theta) = sqrt(2)#