Question
If \( \sin \theta=\frac{\sqrt{7}}{4} \) and \( \cos \theta=\frac{3}{4} \), what is \( \tan \theta \) ? \( \begin{array}{llll}\text { 1) } \frac{4}{3} & \text { 2) } \frac{\sqrt{7}}{4} & \text { 3) } \frac{\sqrt{7}}{3} & \text { 4) }-\frac{\sqrt{7}}{3}\end{array} \)
Ask by Savage Powell. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \tan \theta = \frac{\sqrt{7}}{3} \)
Solution
To find \( \tan \theta \) given \( \sin \theta = \frac{\sqrt{7}}{4} \) and \( \cos \theta = \frac{3}{4} \), you can use the definition of tangent in terms of sine and cosine:
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
Substitute the given values:
\[
\tan \theta = \frac{\frac{\sqrt{7}}{4}}{\frac{3}{4}} = \frac{\sqrt{7}}{3}
\]
So, the correct answer is:
**3) \( \frac{\sqrt{7}}{3} \)**
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Simplify this solution Extra Insights
To find \( \tan \theta \), we use the relationship that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Given \( \sin \theta = \frac{\sqrt{7}}{4} \) and \( \cos \theta = \frac{3}{4} \), we can substitute these values into the formula: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{7}}{4}}{\frac{3}{4}} = \frac{\sqrt{7}}{4} \cdot \frac{4}{3} = \frac{\sqrt{7}}{3} \] Thus, the answer is \( \frac{\sqrt{7}}{3} \), which corresponds to option 3).
