Question

Find the value of each variable.​

Answer 1

Answer:

see explanation

Step-by-step explanation:

Using the exact values of the trigonometric ratios

sin60° = $\frac{\sqrt{3} }{2}$, cos60° = $\frac{1}{2}$

sin45° = cos45° = $\frac{1}{\sqrt{2} }$

Using the sine ratio on the right triangle on the left

sin60° = $\frac{opposite}{hypotenuse}$ = $\frac{a}{4\sqrt{3} }$ = $\frac{\sqrt{3} }{2}$

Cross- multiply

2a = 4$\sqrt{3}$ × $\sqrt{3}$ = 12 ( divide both sides by 2 )

a = 6

Using the cosine ratio on the same right triangle

cos60° = $\frac{adjacent}{hypotenuse}$ = $\frac{c}{4\sqrt{3} }$ = $\frac{1}{2}$

Cross- multiply

2c = 4$\sqrt{3}$ ( divide both sides by 2 )

c = 2$\sqrt{3}$

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Using the sine/cosine ratios on the right triangle on the right

sin45° = $\frac{a}{b}$ = $\frac{6}{b}$ = $\frac{1}{\sqrt{2} }$

Cross- multiply

b = 6$\sqrt{2}$

cos45° = $\frac{d}{b}$ = $\frac{d}{6\sqrt{2} }$ = $\frac{1}{\sqrt{2} }$

Cross- multiply

$\sqrt{2}$ d = 6$\sqrt{2}$ ( divide both sides by $\sqrt{2}$ )

d = 6

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a = 6, b = 6$\sqrt{2}$, c = 2$\sqrt{3}$, d = 6