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

To solve the problem, we start with the given information: \(\sin x = -\frac{3}{5}\), where \(\pi < x < \frac{3\pi}{2}\). This range indicates that \(x\) is in the third quadrant, where sine is negative, and both cosine and tangent are also negative.

1. Using the Pythagorean identity, we know:

\(\sin^2 x + \cos^2 x = 1\)

1. Substituting the value of \(\sin x\), we have:

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

1. Simplifying, we get:

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

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

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

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

1. Next, we find \(\tan x\):

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

\(\tan x = \frac{3}{4}\)

1. Compute \(\tan^2 x - \cos x\):

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

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

1. To combine these fractions, find a common denominator 80:

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

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

1. Finally, compute \(80(\tan^2 x - \cos x)\):

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

1. Therefore, the answer is 109.

This is the correct option among the given choices, justifying that \(80(\tan^2 x - \cos x)\) evaluates to 109 in the specified conditions.

Answer 2

Given:

\[ \sin x = -\frac{3}{5}, \quad \pi < x < \frac{3\pi}{2}. \]

Step 1: Use the Pythagorean identity:

\[ \cos^2 x = 1 - \sin^2 x = 1 - \left(-\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}. \]

Step 2: Determine \( \cos x \):

Since \( \cos x < 0 \) in the third quadrant:

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

Step 3: Calculate \( \tan x \):

\[ \tan x = \frac{\sin x}{\cos x} = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4}. \]

Step 4: Compute \( 80(\tan^2 x - \cos x) \):

\[ \tan^2 x = \left(\frac{3}{4}\right)^2 = \frac{9}{16}, \quad 80\left(\tan^2 x - \cos x\right) = 80\left(\frac{9}{16} - \left(-\frac{4}{5}\right)\right). \]

Step 5: Simplify:

\[ 80\left(\frac{9}{16} + \frac{4}{5}\right) = 80\left(\frac{45}{80} + \frac{64}{80}\right) = 80 \cdot \frac{109}{80} = 109. \]

Final Answer:

\[ \boxed{109.} \]