Question #db4da

2 Answers
Sean
Dec 5, 2017

#theta=pi/6#

Explanation:

.

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

We have an identity that says:

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

Now, we can say:

#sintheta=cos(2theta)#

We also know:

#sintheta=cos(pi/2-theta)#

Therefore,

#cos(pi/2-theta)=cos(2theta)#

#pi/2-theta=2theta#

#pi/2=3theta#

#theta=(pi/2)/3=pi/6#

Nghi N
Dec 5, 2017

Trig identity:
cos^2 t - sin^2 t = cos 2t

Explanation:

Use 2 trig identities:
#cos^2 t - sin^2 t = cos 2t#
#sin t = cos (pi/2 - t)#
Then, we get:
#cos (pi/2 - t) = cos 2t#
Trig unit circle give 2 solutions:
#pi/2 - t = +- 2t#

a. #pi/2 - t = 2t#
#3t = pi/2 + 2kpi#
#t = pi/6 + (2k)pi/3#
b.# pi/2 - t = - 2t#
#t = - pi/2 + 2kpi#
Check.
#t = pi/6# --> #sin t = sin (pi/6) = 1/2# --> #cos 2t = cos pi/3 = 1/2#.
#cos 2t = sin t = 1/2#. Proved
#t = -pi/2 = (3pi)/2# --> #sin t = - 1# --> #cos 2t = cos (3pi) = - 1#
#cos 2t = sin t = - 1#. Proved.
#t = (5pi)/6# (with k = 1) --> #sin t = sin ((5pi)/6) = 1/2# -->
#cos 2t = cos ((10pi)/6) = cos ((5pi)/3) = cos (pi/3) = 1/2#. Proved.

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