Question

What is the length of the side opposite the 30° angle? Explain your reasoning.

Answer 1

Given the triangle ABC as shown below:

The length of the side opposite the 30° angle is evaluated as follows:

Step 1:

Given that the 30° angle is the focus angle, label the sides of the triangle.

Thus,

`\begin{gathered} \text{where }\theta=30^(\circ) \\ AC\Rightarrow hypotenuse\text{ (the longest side of the triangle)} \\ AB\Rightarrow opposite\text{ (the side opposite the focus angle)} \\ BC\Rightarrow adjacent \\ \text{thus, } \\ AC\text{ = 44} \\ AB\text{ = x (unknown length)} \end{gathered}`

Step 2:

Evaluate the unknown side using trignometric ratios.

By trigonometric ratios,

`\begin{gathered} \sin \theta\text{ = }(opposite)/(hypotenuse)=(AB)/(AC) \\ \cos \text{ }\theta\text{ = }(adjacent)/(hyptenuse)=(BC)/(AC) \\ \tan \text{ }\theta\text{ = }(opposite)/(adjacent)=(AB)/(BC) \end{gathered}`

From the above trigonometric ratios, sine θ is used to evaluate the value of the unknown side.

This because the sine θ gives the ralationship between the hypotenuse and the unknown side of the triangle.

Thus,

`\begin{gathered} \sin \theta\text{ = }(opposite)/(hypotenuse)=(AB)/(AC) \\ AB\text{ = x} \\ AC\text{ = 44} \\ \theta\text{ = 30} \\ \Rightarrow\Rightarrow\sin 30\text{ = }(x)/(44) \\ 0.5\text{ = }(x)/(44) \\ \Rightarrow x\text{ = 0.5}*44 \\ x\text{ = 22} \end{gathered}`

Hence, the value of the unknown side is 22.