Find the Values of Other Five Trigonometric Functions if cot x = 12/5, x in Quadrant III
Question:
Find the values of other five trigonometric functions in the following : \[ \cot x = \frac{12}{5}, \quad x \text{ in quadrant III} \]
Solution
Given,
\[ \cot x = \frac{12}{5} \]
Since
\[ \cot x = \frac{\text{Base}}{\text{Perpendicular}} \]
Therefore,
\[ \text{Base} = 12, \quad \text{Perpendicular} = 5 \]
Using Pythagoras theorem,
\[ \text{Hypotenuse} = \sqrt{12^2 + 5^2} \]
\[ = \sqrt{144 + 25} \]
\[ = \sqrt{169} \]
\[ = 13 \]
In quadrant III, sine and cosine are negative while tangent and cotangent are positive.
Now find the other five trigonometric functions
\[ \sin x = \frac{-5}{13} \]
\[ \cos x = \frac{-12}{13} \]
\[ \tan x = \frac{5}{12} \]
\[ \sec x = \frac{-13}{12} \]
\[ \csc x = \frac{-13}{5} \]
Final Answer
\[ \sin x = \frac{-5}{13} \]
\[ \cos x = \frac{-12}{13} \]
\[ \tan x = \frac{5}{12} \]
\[ \sec x = \frac{-13}{12} \]
\[ \csc x = \frac{-13}{5} \]