If 3π/2≤ θ≤2π and cosθ=8/13, find sin(2θ), cos(2θ), and tan (2θ)?

Question

How can you use double angle formulas to solve this problem?

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Answer 1 · 2017-12-12T01:10:51

Please see below.

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We know:

#sin^2x+cos^2x=1#, therefore, #sinx=sqrt(1-cos^2x)#

#sintheta=+-sqrt(1-(8/13)^2)=+-sqrt(1-64/169)=+-sqrt((169-64)/169)#

#sintheta=+-sqrt(105/169)=+-sqrt105/13#

But because #theta# is between #(3pi)/2# and #2pi#, i.e in the third quadrant, #sintheta=-sqrt105/13#

#sin2theta=2sinthetacostheta=2(sqrt105/13)(8/13)=(16sqrt105)/169#

#cos2theta=1-2sin^2theta=1-2(105/169)=(169-210)/169=-41/169#

#tan2theta=(sin2theta)/(cos2theta)=((16sqrt105)/169)/(-41/169)=-(16sqrt105)/41#