Answer
The ratio of the side opposite to the acute angle to the hypotenuse of the right triangle is called the sine of an acute angle of a right triangle. The ratio of the side next to the angle and the hypotenuse of the right triangle is known as the cosine of an acute angle of a right triangle.
Work Step by Step
The longest side of the right triangle is known as the hypotenuse. There isan acute angle and a right angle in right angle triangle. The two legs of the triangle are related to an acute angle. One leg is known as side opposite to the acute angle and other leg is side adjacent to the acute angle.
Example:
It is required to compute remaining measures of the triangle.If the length of one side, i.e., the hypotenuse and the measure of the acute angles are given, use the trigonometry to determine the measures of the triangle lengths.
The length of one side is \[10\text{ ft}\]and the measure of an acute angle is\[36{}^\circ \].The measure of another acute angle can be computed as follows:
\[90{}^\circ -36{}^\circ =54{}^\circ \]
The side length opposite to the acute angle can be measured as using sine ratio:
\[\begin{align}
& \sin \theta =\frac{a}{10\text{ ft}} \\
& \sin 36{}^\circ =\frac{a}{10\text{ ft}} \\
& a=10\times \sin 36{}^\circ \\
& =10\times 0.588
\end{align}\]
\[=5.88ft\]
Compute the length of the side next to the acute angle using cosine ratio as follows:
\[\begin{align}
& cos\theta =\frac{b}{10ft} \\
& cos36{}^\circ =\frac{b}{10ft} \\
& b=10\times cos36{}^\circ \\
& =10\times 0.809 \\
& =8.09ft
\end{align}\]
Hence, the length a is\[5.88\text{ ft}\]and b is\[8.09\text{ ft}\].
