Question

A/Sin45=b/Sin60=c/Sin75
find
a:b:c​

Answer 1

4 : 2√6 : √3 + 1

Explanation:

Our equation is:

`(a)/(sin(45)) =(b)/(sin(60)) =(c)/(sin(75))`

These are a few trig identities that we want to memorize:

- sin(45) = √2/2

- sin(60) = √3/2

- sin(75) = sin(45 + 30) = sin(45) * cos(30) + sin(30) * cos(45) = (√2/2)(√3/2) + (1/2)(√2/2) = √6/4 + √2/4 = (√6 + √2)/4

Put these in:

`(a)/(sin(45)) =(b)/(sin(60)) =(c)/(sin(75))`

`(a)/(√(2)/2) =(b)/(√(3) /2) =(c)/((√(6) +√(2) )/(4) )`

`(2)/(√(2) ) a=(2)/(√(3) ) b=(4)/(√(6)+√(2) ) c`

`√(2)a=(2√(3) )/(3) b=(√(6)-√(2))c`

`a=(2√(3) )/(3) *(√(6) -√(2) )=(6√(2) -2√(6) )/(3)`

`b=√(2) *(√(6) -√(2))=2√(3) -2`

`c=√(2) *(2√(3) )/(3) =(2√(6) )/(3)`

Then the ratio of a:b:c is:

(6√2 - 2√6) : (6√3 - 6) : (2√6), which simplifies to:

4 : 2√6 : √3 + 1

If you wanted, you could use a calculator to find the decimal form of that.

Answer 2

that is the solution to the question