Find the Values of Other Five Trigonometric Functions

Question:

Find the values of other five trigonometric functions in the following :

\[ \sin x = \frac{3}{5}, \quad x \text{ lies in Quadrant I} \]

Solution

Given,

\[ \sin x = \frac{3}{5} \]

Since \(x\) lies in Quadrant I, all trigonometric functions are positive.

Using the relation:

\[ \sin x = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]

Take,

\[ \text{Perpendicular} = 3, \quad \text{Hypotenuse} = 5 \]

Using Pythagoras theorem:

\[ \text{Base} = \sqrt{5^2 – 3^2} \]

\[ = \sqrt{25 – 9} \]

\[ = \sqrt{16} = 4 \]

Now find the remaining trigonometric functions.

\[ \cos x = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{4}{5} \]

\[ \tan x = \frac{\text{Perpendicular}}{\text{Base}} = \frac{3}{4} \]

\[ \csc x = \frac{1}{\sin x} = \frac{5}{3} \]

\[ \sec x = \frac{1}{\cos x} = \frac{5}{4} \]

\[ \cot x = \frac{1}{\tan x} = \frac{4}{3} \]

\[ \cos x = \frac{4}{5} \]

\[ \tan x = \frac{3}{4} \]

\[ \csc x = \frac{5}{3} \]

\[ \sec x = \frac{5}{4} \]

\[ \cot x = \frac{4}{3} \]

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