Question

Find fraction value of cos 72 degree using complex numbers.

Answer 1

Step-by-step explanation:

follow the steps of the answer

Answer 2

Correct Question:

Find fraction value of cos 72 degree using complex numbers.$\:$

Explanation:

Approach:

Note that 5⋅72=360 . Thus, if we denote by φ the angle, we have

Let's begin:-

(cosφ+isinφ)5=1

Expanding the power, we obtain

cos5φ+5icos4φsinφ−10cos3φsin2φ−10icos2φsin3φ+5cosφsin4φ+isin5φ=1

Therefore, equating the imaginary part to 0, we get

=>5cos4φsinφ−10cos2φsin3φ+sin5φ=0

Since sinφ≠0 we can remove the factor; then setting x=cosφ and recalling that sin2φ=1−cos2φ=1−x2 we obtain

=>5x4−10x2(1−x2)+(1−x2)2=0

that simplifies to

=>16x4–12x2+1=0

leading to

=>x2=6+20−−√16=5+25–√+116

=>(5–√+1)242

or

=>x2=6−20−−√16=5−25–√+116

=>(5–√−1)242

We know that x=cosφ>0 , so we have either

=>x=5–√+14

or

=>x=5–√−14

As the latter is less than 1/2=cos(π/3)=cos60∘ it is the one we were looking for:

cos72∘=5–√−14

Hence,◎Verified◎