Answer
Value of \[\sin A=\frac{4}{5}\], \[\cos A=\frac{3}{5}\], and \[\tan A=\frac{4}{3}\].
Work Step by Step
The length of hypotenuse that is \[\left( c=35 \right)\], Side adjacent to acute angle A that is \[\left( b=21 \right)\]. First, compute side opposite to acute angle A that is a, using the Pythagorean Theorem.
Compute the side opposite to acute angle A that is a; of a triangle, by the below mentioned formula:
\[\begin{align}
& c\text{ }=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
& \text{35 }=\sqrt{{{a}^{2}}+{{21}^{2}}} \\
& {{35}^{2}}={{a}^{2}}+{{21}^{2}} \\
& 1225={{a}^{2}}+441
\end{align}\]
Further it can be simplified as:
\[\begin{align}
& {{a}^{2}}=1225-441 \\
& {{a}^{2}}=784
\end{align}\]
Take square root both the sides as follows:
\[\begin{align}
& \sqrt{{{a}^{2}}}=\sqrt{784} \\
& a=28
\end{align}\]
Now, compute the different trigonometric ratios using the formulas:
\[\begin{align}
& \sin \text{ }A=\frac{\text{Side opposite to angle }A}{\text{Hypotenuse}} \\
& =\frac{28}{35} \\
& =\frac{4}{5}
\end{align}\]
Therefore, \[\sin A=\frac{4}{5}\]
\[\begin{align}
& \cos \text{ }A=\frac{\text{Side adjacent to angle }A}{\text{Hypotenuse}} \\
& =\frac{21}{35} \\
& =\frac{3}{5}
\end{align}\]
Therefore, \[\cos A=\frac{3}{5}\]
\[\begin{align}
& \text{tan }A=\frac{\text{Side opposite angle }A}{\text{Side adjacent to angle }A} \\
& =\frac{28}{21} \\
& =\frac{4}{3}
\end{align}\]
Therefore, \[\tan A=\frac{4}{3}\]
Hence, the value of trigonometric ratios is, \[\sin A=\frac{4}{5}\], \[\cos A=\frac{3}{5}\], and \[\tan A=\frac{4}{3}\].
