Question

Determine the value of each variable in the figure below. Keep answers in simplest

rodicol form
c
42
30°
609

Answer 1

`\begin{gathered} \text{d = 21 }\sqrt[]{3} \\ a\text{ = 21} \\ c\text{ = 42 }\sqrt[]{3} \\ b\text{ = 63} \end{gathered}`Step-by-step explanation:

The triangles is the bigger triangle are both right angled

Considering the triangle by the left:

when angle = 60°

opposite = d

hypotenuse = 42

To get d, we will use sine ratio:

`\begin{gathered} \sin \text{ 60}\degree\text{ = }(opposite)/(hypotenuse) \\ \sin \text{ 60}\degree\text{= }(d)/(42) \\ d\text{ = 42 }*\text{ sin 60}\degree\text{ } \\ In\text{ radical form sin60}\degree\text{ = }\frac{\sqrt[]{3}}{2} \\ d\text{ =42 }*\frac{\sqrt[]{3}}{2} \\ d\text{ = }21\text{ }\sqrt[]{3} \end{gathered}`

To get a, we will use cosine ratio:

`\begin{gathered} \cos \text{ 60}\degree\text{ = }(adjacent)/(hypotenuse) \\ \cos \text{ 60}\degree\text{in radical form = }(1)/(2) \\ \text{adjacent = a} \\ \cos \text{ 60}\degree\text{ = }(a)/(42) \\ a\text{ = 42(}\cos \text{ 60}\degree)\text{ }=\text{ 42(}(1)/(2)) \\ a\text{ = }21 \end{gathered}`

Considering the triangle by the right:

opposite = d = 21√3

angle = 30°

hypotenuse = c

To get c, we will use sine ratio:

`\begin{gathered} \sin \text{ 30}\degree\text{ = }(opposite)/(hypotenuse) \\ \sin \text{ 30}\degree\text{ = }(21\surd3)/(c) \\ c\text{ = }\frac{21\surd3}{\sin \text{ 30}\degree}\text{ ; }\sin \text{ 30}\degree\text{ = 1/2 } \\ c\text{ = }(21\surd3)/((1)/(2))=\text{ }21\surd3\text{ }*\text{ 2} \\ c\text{ = 42 }\sqrt[]{3} \end{gathered}`

b = adjacent

To get b, we will use cosine ratio:

`\begin{gathered} \cos \text{ 30}\degree\text{ = }\frac{adjacent}{\text{hypotenuse}}\text{= }(b)/(c) \\ \cos \text{ 30}\degree\text{ }=\frac{b}{42\text{ }\sqrt[]{3}} \\ \cos \text{ 30}\degree\text{ in radical form = }\frac{\sqrt[]{3}}{2} \\ b\text{ = }42\text{ }\sqrt[]{3}\text{ }*\text{ }\cos \text{ 30}\degree\text{ = }42\text{ }\sqrt[]{3}\text{ }*\frac{\sqrt[]{3}}{2} \\ b\text{ = }(42(3))/(2) \\ b\text{ = 63} \end{gathered}`