\(\displaystyle{4}{{\sin{{72}}}^{\circ}{{\sin{{36}}}^{\circ}=}}\sqrt{{{5}}}\)

ikramkeyslo4s

ikramkeyslo4s

Answered question

2022-03-31

4sin72sin36=5

Answer & Explanation

crazyrocketrz5z

crazyrocketrz5z

Beginner2022-04-01Added 11 answers

A key observation is 36+72+72=180. This means
sin72=sin(36+72)=sin36cos72+cos36sin72
Using the identity sin72=2sin36cos36, we get
2sin36cos36=sin36cos72+2sin36cos236
Now notice sin36 appears everywhere, so we can divide by it...
2cos36=cos72+2cos236
Use the identity cos72=2cos2361, we get
2cos36=4cos2361
So cos36 is one of the solutions to 2x=4x21. Using the quadratic formula, you see that the solutions are
2±204=1±52
But you know that cos36 must be positive, so you know the correct value. Now you can get sin72 from any identity you wish, and multiply.

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