Question

Find the value of each trigonometric ratio

Answer 1

The value of each trigonometric ratio is

$\sin A=\frac{15}{17}, \ \cos A=\frac{8}{17}, \ \tan A =\frac{15}{8}$

$\csc A=\frac{17}{15}, \ \sec A=\frac{17}{8}, \ \cot A =\frac{8}{15}$

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,

$\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}$

$\sin A=\frac{BC}{AC}$

$\sin A=\frac{30}{34}=\frac{15}{17}$

$\cos \theta=\frac{\text { adjacent }}{\text { hypotenuse }}$

$\cos A=\frac{AB}{AC}$

$\cos A=\frac{16}{34}=\frac{8}{17}$

$\tan \theta=\frac{\text { opposite }}{\text { adjacent }}$

$\tan A=\frac{BC}{AB}$

$\tan A=\frac{30}{16}=\frac{15}{8}$

$\csc A =\frac{1}{\sin A}$

$\csc A =\frac{17}{15}$

$\sec A =\frac{1}{\cos A}$

$\sec A =\frac{17}{8}$

$\cot A =\frac{1}{\tan A}$

$\cot A =\frac{8}{15}$

Hence the value of each trigonometric ratio is

$\sin A=\frac{15}{17}, \ \cos A=\frac{8}{17}, \ \tan A =\frac{15}{8}$

$\csc A=\frac{17}{15}, \ \sec A=\frac{17}{8}, \ \cot A =\frac{8}{15}$