Question

If $\sin x = -\frac{3}{5}$, where $\pi<x<\frac{3\pi}{2}$, then $80(\tan^2 x - \cos x)$ is equal to:

• A. 109
• B. 108
• C. 18
• D. 19

Answer 1

The Correct Option is A

Given: \( \sin x = -\frac{3}{5} \) and \( \pi<x<\frac{3\pi}{2} \). This implies \( x \) is in the third quadrant, where sine, cosine, and tangent are negative.

1. Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \):

\(\left(-\frac{3}{5}\right)^2 + \cos^2 x = 1\)

\(\frac{9}{25} + \cos^2 x = 1\)

\(\cos^2 x = 1 - \frac{9}{25} = \frac{16}{25}\)

\(\cos x = -\frac{4}{5}\) (since cosine is negative in the third quadrant)

1. Calculate \( \tan x \):

\(\tan x = \frac{\sin x}{\cos x} = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4}\)

1. Evaluate \( \tan^2 x - \cos x \):

\(\tan^2 x = \left(\frac{3}{4}\right)^2 = \frac{9}{16}\)

\(\tan^2 x - \cos x = \frac{9}{16} - \left(-\frac{4}{5}\right)\)

1. Find a common denominator (80) to combine the fractions:

\(\frac{9}{16} = \frac{45}{80}\) and \(-\frac{4}{5} = -\frac{64}{80}\)

\(\tan^2 x - \cos x = \frac{45}{80} + \frac{64}{80} = \frac{109}{80}\)

1. Compute \( 80(\tan^2 x - \cos x) \):

\(80 \times \frac{109}{80} = 109\)

1. The result is 109.

This confirms that \( 80(\tan^2 x - \cos x) \) equals 109 under the given conditions.