How do you find all solutions of #2cos^2(θ)-1=0# to be #pi/4 +kpi# and #3pi/4 + kpi#? I keep getting #2kpi# instead of #kpi#
2#cos^2# θ-1=0
2#cos^2# θ=1
#cos^2# θ=#1/2#
cosθ=1/#sqrt2# =#sqrt2# /2
θ=±#pi# /4 +2k#pi#
θ=#pi# /4 +2#pi# k
or
θ=3#pi# /4 +2#pi# k
but my book says the answer is:
θ=#pi# /4 +k#pi#
or
θ=3#pi# /4 +k#pi#
2
2
cosθ=1/
θ=±
θ=
or
θ=3
but my book says the answer is:
θ=
or
θ=3
1 Answer
Checking our unit circle, we find that
The problem with your given answer is that it does not account for solutions at which the angle is equivalent to
Noting that
Similarly, we can combine
The book stops there, but if we wanted to put the answer in its most concise form, we could just note that every answer is an integer number of rotations of