Answer
Value of \[\sin A=\frac{24}{25}\], \[\cos A=\frac{7}{25}\], and \[\tan A=\frac{24}{7}\].
Work Step by Step
The length of hypotenuse that is \[\left( c=25 \right)\], the side opposite to acute angle A that is \[\left( a=24 \right)\]. First, it is required to compute side adjacent to acute angle A, that is b, using the Pythagorean Theorem.
Compute the side adjacent to acute angle A; that is b of a triangle, by the formula as follows:
\[\begin{align}
& c\text{ }=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
& 25=\sqrt{{{24}^{2}}+{{b}^{2}}} \\
& {{25}^{2}}={{24}^{2}}+{{b}^{2}} \\
& 625=576+{{b}^{2}}
\end{align}\]
Further it can be simplified as:
\[\begin{align}
& {{b}^{2}}=625-576 \\
& {{b}^{2}}=49
\end{align}\]
Take square root both the sides as follows:
\[\begin{align}
& \sqrt{{{b}^{2}}}=\sqrt{49} \\
& \text{b}=7
\end{align}\]
Now, compute the different trigonometric ratios using the formulas:
\[\begin{align}
& \sin A=\frac{\text{Side opposite to angle }A}{\text{Hypotenuse}} \\
& =\frac{24}{25}
\end{align}\]
Therefore, \[\sin A=\frac{24}{25}\]
\[\begin{align}
& \cos \text{ }A=\frac{\text{Side adjacent to angle }A}{\text{Hypotenuse}} \\
& =\frac{7}{25}
\end{align}\]
Therefore, \[\cos A=\frac{7}{25}\]
\[\begin{align}
& \text{tan }A=\frac{\text{Side opposite angle A}}{\text{Side adjacent to angle A}} \\
& =\frac{24}{7}
\end{align}\]
Therefore, \[\tan A=\frac{24}{7}\]
Hence, the value of trigonometric ratios is, \[\sin A=\frac{24}{25}\], \[\cos A=\frac{7}{25}\], and \[\tan A=\frac{24}{7}\].
