Answer
Value of \[\sin A=\frac{5}{13}\], \[\cos A=\frac{12}{13}\], and \[\tan A=\frac{5}{12}\].
Work Step by Step
Length of hypotenuse that is \[\left( c=26 \right)\], Side opposite to acute angle A that is \[\left( a=10 \right)\]. Firstly, it is required to compute side adjacent to acute angle A that is b using the Pythagorean Theorem as follows:
The side adjacent to acute angle A, that is b of a triangle, is given by the formula:
\[\begin{align}
& c\text{ }=\sqrt{{{a}^{2}}+{{b}^{2}}} \\
& \text{26 }=\sqrt{{{10}^{2}}+{{b}^{2}}} \\
& {{26}^{2}}={{10}^{2}}+{{b}^{2}} \\
& 676=100+{{b}^{2}}
\end{align}\]
Further it can be simplified as:
\[\begin{align}
& {{b}^{2}}=676-100 \\
& {{b}^{2}}=576
\end{align}\]
Take square root both the sides as follows:
\[\begin{align}
& {{b}^{2}}=\sqrt{576} \\
& \text{b }=24
\end{align}\]
Now, compute the different trigonometric ratios using the formulas:
\[\begin{align}
& \sin A=\frac{\text{Side opposite to angle }A}{\text{Hypotenuse}} \\
& =\frac{10}{26} \\
& =\frac{5}{13}
\end{align}\]
Therefore, \[\sin A=\frac{5}{13}\]
\[\begin{align}
& \cos A=\frac{\text{Side adjacent to angle }A}{\text{Hypotenuse}} \\
& =\frac{24}{26} \\
& =\frac{12}{13}
\end{align}\]
Therefore, \[\cos A=\frac{12}{13}\]
\[\begin{align}
& \text{tan }A=\frac{\text{Side opposite angle }A}{\text{Side adjacent to angle }A} \\
& =\frac{10}{24} \\
& =\frac{5}{12}
\end{align}\]
Therefore, \[\tan A=\frac{5}{12}\]
Hence, the value of trigonometric ratios is,
\[\sin A=\frac{5}{13}\], \[\cos A=\frac{12}{13}\], and \[\tan A=\frac{5}{12}\].
