Answer:
see explanation
Step-by-step explanation:
Using the exact values of the trigonometric ratios
sin60° = [tex]\frac{\sqrt{3} }{2}[/tex], cos60° = [tex]\frac{1}{2}[/tex]
sin45° = cos45° = [tex]\frac{1}{\sqrt{2} }[/tex]
Using the sine ratio on the right triangle on the left
sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{a}{4\sqrt{3} }[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex]
Cross- multiply
2a = 4[tex]\sqrt{3}[/tex] × [tex]\sqrt{3}[/tex] = 12 ( divide both sides by 2 )
a = 6
Using the cosine ratio on the same right triangle
cos60° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{c}{4\sqrt{3} }[/tex] = [tex]\frac{1}{2}[/tex]
Cross- multiply
2c = 4[tex]\sqrt{3}[/tex] ( divide both sides by 2 )
c = 2[tex]\sqrt{3}[/tex]
------------------------------------------------------------------------------------------
Using the sine/cosine ratios on the right triangle on the right
sin45° = [tex]\frac{a}{b}[/tex] = [tex]\frac{6}{b}[/tex] = [tex]\frac{1}{\sqrt{2} }[/tex]
Cross- multiply
b = 6[tex]\sqrt{2}[/tex]
cos45° = [tex]\frac{d}{b}[/tex] = [tex]\frac{d}{6\sqrt{2} }[/tex] = [tex]\frac{1}{\sqrt{2} }[/tex]
Cross- multiply
[tex]\sqrt{2}[/tex] d = 6[tex]\sqrt{2}[/tex] ( divide both sides by [tex]\sqrt{2}[/tex] )
d = 6
------------------------------------------------------------------------------------------------
a = 6, b = 6[tex]\sqrt{2}[/tex], c = 2[tex]\sqrt{3}[/tex], d = 6
