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,
[tex]\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}[/tex]
Step 2:
Evaluate the unknown side using trignometric ratios.
By trigonometric ratios,
[tex]\begin{gathered} \sin \theta\text{ = }\frac{opposite}{hypotenuse}=\frac{AB}{AC} \\ \cos \text{ }\theta\text{ = }\frac{adjacent}{hyptenuse}=\frac{BC}{AC} \\ \tan \text{ }\theta\text{ = }\frac{opposite}{adjacent}=\frac{AB}{BC} \end{gathered}[/tex]
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,
[tex]\begin{gathered} \sin \theta\text{ = }\frac{opposite}{hypotenuse}=\frac{AB}{AC} \\ AB\text{ = x} \\ AC\text{ = 44} \\ \theta\text{ = 30} \\ \Rightarrow\Rightarrow\sin 30\text{ = }\frac{x}{44} \\ 0.5\text{ = }\frac{x}{44} \\ \Rightarrow x\text{ = 0.5}\times44 \\ x\text{ = 22} \end{gathered}[/tex]
Hence, the value of the unknown side is 22.
