The value of x satisfying the equation

Gustavo Lam

Gustavo Lam

Answered question

2022-04-09

The value of x satisfying the equation x=2+22+x
(A) 2cos10
(B) 2cos20
(C) 2cos40
(D) 2cos80

Answer & Explanation

lildeutsch11xq2j

lildeutsch11xq2j

Beginner2022-04-10Added 13 answers

Using the trigonometric identity,
cos2x=2cos2x1=12sin2x
2cos2x+2=4cos2x
We are looking for an angle that allows for the relationship sinx=cos(π2x) in the final step. This is because, working from the inside out, the first gives a cosine, the second gives a sine, so in order for the last one to be a cosine, we must change the sine into a cosine through the above relation.
A simple check/use of intuition finds that C is the answer.
x=2+22+x
If x=2cos40
2+22+2cos40=2+24cos220
=2+22cos20=2+4sin210=2+2sin10
=2+2cos80=2cos40=x

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