Question

How do you find the exact value of `sin 84` using the sum and difference, double angle or half angle formulas?

Answer 1

`sin84^@=1/8(sqrt15+sqrt3+sqrt(10-2sqrt5))`

Explanation:

Let `A=36^@`, hence `5A=180^@` and

`2A=180^@-3A` and `sin2A=sin(180^@-3A)=sin3A`

or `2sinAcosA=3sinA-4sin^3A` and as `sinA!=0`, dividing by it we get

`2cosA=3-4sin^2A` and `2cosA=3-4(1-cos^2A)`

i.e. `4cos^2A-2cosA-1=0`

and using quadratic formula `cosA=(2+-sqrt(2^2-4xx4xx(-1)))/8`

= `(2+-sqrt20)/8=(sqrt5+1)/4`

As `cos36^@=sin54^@=(sqrt5+1)/4`

and `cos54^@=sqrt(1-(sqrt5+1)^2/16)=1/4sqrt(16-(5+1+2sqrt5)`

= `sqrt(10-2sqrt5)/4`

`sin84^@=sin(54^@+30^@)`

= `sin54^@cos30^@+cos54^@sin30^@`

= `(sqrt5+1)/4xxsqrt3/2+sqrt(10-2sqrt5)/4xx1/2`

= `1/8(sqrt15+sqrt3+sqrt(10-2sqrt5))`