Find sin x and cos x

Question:

\[ \sin x + \cos x = 0 \]

and \(x\) lies in the fourth quadrant, find the values of:

\[ \sin x \quad \text{and} \quad \cos x \]

Solution

Given,

\[ \sin x + \cos x = 0 \]

Therefore,

\[ \sin x = -\cos x \]

Squaring both sides:

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

Using the identity:

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

Substitute:

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

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

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

\[ \cos x = \pm \frac{1}{\sqrt{2}} \]

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

Hence,

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

and

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

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

\[ \cos x = \frac{\sqrt{2}}{2} \]

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