Question

Find tan θ, given that sine theta equals negative Start Fraction 4 over 5 End Fraction and cosine theta greater-than 0.
(1 point)
Responses

negative Start Fraction 5 over 3 End Fraction
Image with alt text: negative Start Fraction 5 over 3 End Fraction

negative Start Fraction 4 over 3 End Fraction
Image with alt text: negative Start Fraction 4 over 3 End Fraction

Start Fraction 3 over 5 End Fraction
Image with alt text: Start Fraction 3 over 5 End Fraction

Start Fraction 5 over 4 End Fraction

Answer 1

To find \(\tan \theta\) given that \(\sin \theta = -\frac{4}{5}\) and \(\cos \theta > 0\), we can follow these steps:

1. **Identify the quadrant**: Since \(\sin \theta\) is negative and \(\cos \theta\) is positive, \(\theta\) must be in the fourth quadrant.

2. **Use the Pythagorean identity**: Recall that \(\sin^2 \theta + \cos^2 \theta = 1\).

\[
\sin^2 \theta = \left(-\frac{4}{5}\right)^2 = \frac{16}{25}
\]
\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{16}{25} = \frac{9}{25}
\]

3. **Calculate \(\cos \theta\)**: Since \(\cos \theta > 0\) in the fourth quadrant, we take the positive square root:

\[
\cos \theta = \sqrt{\frac{9}{25}} = \frac{3}{5}
\]

4. **Calculate \(\tan \theta\)**: The tangent function is defined as:

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

Thus, the value of \(\tan \theta\) is

\[
\boxed{-\frac{4}{3}}
\]