Find the Values of Other Five Trigonometric Functions

Question:

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

\[ \tan x = \frac{3}{4}, \quad x \text{ lies in Quadrant III} \]

Solution

Given,

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

Since \(x\) lies in Quadrant III, both sine and cosine are negative while tangent is positive.

Using the relation:

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

Take,

\[ \text{Perpendicular} = 3, \quad \text{Base} = 4 \]

Using Pythagoras theorem:

\[ \text{Hypotenuse} = \sqrt{3^2 + 4^2} \]

\[ = \sqrt{9 + 16} \]

\[ = \sqrt{25} = 5 \]

In Quadrant III:

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

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

Now find the remaining trigonometric functions.

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

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

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

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

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

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

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

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

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