Answer
Value of \[\sin A=\frac{9}{41}\], \[\cos A=\frac{40}{41}\], and \[\tan A=\frac{9}{40}\].
Work Step by Step
The length of hypotenuse that is \[\left( c=41 \right)\], Side adjacent to acute angle A that is \[\left( b=40 \right)\]. Firstly, compute side opposite to acute angle A\[\left( a \right)\], using the Pythagorean Theorem. The side opposite to acute angle A\[\left( a \right)\]of a triangle is given by the formula:
\[\begin{align}
& c\text{ }=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
& \text{41 }=\sqrt{{{a}^{2}}+{{40}^{2}}} \\
& {{41}^{2}}={{a}^{2}}+{{40}^{2}} \\
& 1681={{a}^{2}}+1600
\end{align}\]
Further it can be simplified as:
\[\begin{align}
& {{a}^{2}}=1681-1600 \\
& {{a}^{2}}=81
\end{align}\]
Take square root both the sides as follows:
\[\begin{align}
& \sqrt{{{a}^{2}}}=\sqrt{81} \\
& a\text{ }=9
\end{align}\]
Now, compute the different trigonometric ratios using the formulas:
\[\begin{align}
& \sin \text{ }A=\frac{\text{Side opposite to angle }A}{\text{Hypotenuse}} \\
& =\frac{9}{41}
\end{align}\]
Therefore, \[\sin A=\frac{9}{41}\]
\[\begin{align}
& \cos \text{ }A=\frac{\text{Side adjacent to angle }A}{\text{Hypotenuse}} \\
& =\frac{40}{41}
\end{align}\]
Therefore, \[\cos A=\frac{40}{41}\]
\[\begin{align}
& \text{tan }A=\frac{\text{Side opposite angle }A}{\text{Side adjacent to angle }A} \\
& =\frac{9}{40}
\end{align}\]
Therefore, \[\tan A=\frac{9}{40}\]
Hence, the value of trigonometric ratios is, \[\sin A=\frac{9}{41}\], \[\cos A=\frac{40}{41}\], and \[\tan A=\frac{9}{40}\].
