Find the Values of Other Five Trigonometric Functions

Question:

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

\[ \cos x = -\frac{1}{2}, \quad x \text{ lies in Quadrant II} \]

Solution

Given,

\[ \cos x = -\frac{1}{2} \]

Since \(x\) lies in Quadrant II, sine is positive and cosine is negative.

Using the identity:

\[ \sin^2 x + \cos^2 x = 1 \]

Substitute the value of \(\cos x\):

\[ \sin^2 x + \left(-\frac{1}{2}\right)^2 = 1 \]

\[ \sin^2 x + \frac{1}{4} = 1 \]

\[ \sin^2 x = 1 – \frac{1}{4} \]

\[ \sin^2 x = \frac{3}{4} \]

\[ \sin x = \frac{\sqrt{3}}{2} \]

Now find the remaining trigonometric functions.

\[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \]

\[ \sec x = \frac{1}{\cos x} = \frac{1}{-\frac{1}{2}} = -2 \]

\[ \csc x = \frac{1}{\sin x} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

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

\[ \sin x = \frac{\sqrt{3}}{2} \]

\[ \tan x = -\sqrt{3} \]

\[ \sec x = -2 \]

\[ \csc x = \frac{2\sqrt{3}}{3} \]

\[ \cot x = -\frac{\sqrt{3}}{3} \]

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