The value of each trigonometric ratio is
[tex]$\sin A=\frac{15}{17}, \ \cos A=\frac{8}{17}, \ \tan A =\frac{15}{8}[/tex]
[tex]$\csc A=\frac{17}{15}, \ \sec A=\frac{17}{8}, \ \cot A =\frac{8}{15}[/tex]
Solution:
The given triangle is right triangle.
AC (hypotenuse) = 34, AB (adjacent) = 16, BC (opposite) = 30
To find the trigonometric ratios:
Using trigonometric formulas for right triangle,
[tex]$\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}[/tex]
[tex]$\sin A=\frac{BC}{AC}[/tex]
[tex]$\sin A=\frac{30}{34}=\frac{15}{17}[/tex]
[tex]$\cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }}[/tex]
[tex]$\cos A=\frac{AB}{AC}[/tex]
[tex]$\cos A=\frac{16}{34}=\frac{8}{17}[/tex]
[tex]$\tan \theta=\frac{\text { opposite }}{\text { adjacent }}[/tex]
[tex]$\tan A=\frac{BC}{AB}[/tex]
[tex]$\tan A=\frac{30}{16}=\frac{15}{8}[/tex]
[tex]$\csc A =\frac{1}{\sin A}[/tex]
[tex]$\csc A =\frac{17}{15}[/tex]
[tex]$\sec A =\frac{1}{\cos A}[/tex]
[tex]$\sec A =\frac{17}{8}[/tex]
[tex]$\cot A =\frac{1}{\tan A}[/tex]
[tex]$\cot A =\frac{8}{15}[/tex]
Hence the value of each trigonometric ratio is
[tex]$\sin A=\frac{15}{17}, \ \cos A=\frac{8}{17}, \ \tan A =\frac{15}{8}[/tex]
[tex]$\csc A=\frac{17}{15}, \ \sec A=\frac{17}{8}, \ \cot A =\frac{8}{15}[/tex]
