Answer:
4 : 2√6 : √3 + 1
Step-by-step explanation:
Our equation is:
[tex]\frac{a}{sin(45)} =\frac{b}{sin(60)} =\frac{c}{sin(75)}[/tex]
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:
[tex]\frac{a}{sin(45)} =\frac{b}{sin(60)} =\frac{c}{sin(75)}[/tex]
[tex]\frac{a}{\sqrt{2}/2} =\frac{b}{\sqrt{3} /2} =\frac{c}{\frac{\sqrt{6} +\sqrt{2} }{4} }[/tex]
[tex]\frac{2}{\sqrt{2} } a=\frac{2}{\sqrt{3} } b=\frac{4}{\sqrt{6}+\sqrt{2} } c[/tex]
[tex]\sqrt{2}a=\frac{2\sqrt{3} }{3} b=(\sqrt{6}-\sqrt{2})c[/tex]
[tex]a=\frac{2\sqrt{3} }{3} *(\sqrt{6} -\sqrt{2} )=\frac{6\sqrt{2} -2\sqrt{6} }{3}[/tex]
[tex]b=\sqrt{2} *(\sqrt{6} -\sqrt{2})=2\sqrt{3} -2[/tex]
[tex]c=\sqrt{2} *\frac{2\sqrt{3} }{3} =\frac{2\sqrt{6} }{3}[/tex]
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.
