Question
If \( \csc (t) \approx 1.18 \), and \( \cos (t) \approx 0.53 \), find \( \tan (t) \). Round to 2 decimal places. \( \tan (t) \approx \)
Ask by Washington Campos. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \tan(t) \approx 1.60 \)
Solution
To find \( \tan(t) \) given \( \csc(t) \approx 1.18 \) and \( \cos(t) \approx 0.53 \), follow these steps:
1. **Find \( \sin(t) \) from \( \csc(t) \):**
\[
\csc(t) = \frac{1}{\sin(t)} \implies \sin(t) = \frac{1}{1.18} \approx 0.8475
\]
2. **Use the definition of \( \tan(t) \):**
\[
\tan(t) = \frac{\sin(t)}{\cos(t)} \approx \frac{0.8475}{0.53}
\]
3. **Calculate the value:**
\[
\frac{0.8475}{0.53} \approx 1.597
\]
4. **Round to two decimal places:**
\[
\tan(t) \approx 1.60
\]
**Answer:**
\[
\tan(t) \approx 1.60
\]
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Simplify this solution Extra Insights
To find \( \tan(t) \), we can use the relationship: \[ \tan(t) = \frac{\sin(t)}{\cos(t)}. \] Since \( \csc(t) \) is the reciprocal of \( \sin(t) \), we have: \[ \sin(t) = \frac{1}{\csc(t)} \approx \frac{1}{1.18} \approx 0.8475. \] Now substituting the values we have: \[ \tan(t) = \frac{\sin(t)}{\cos(t)} \approx \frac{0.8475}{0.53}. \] Calculating \( \tan(t) \): \[ \tan(t) \approx 1.60. \] So, when rounded to two decimal places, we find: \[ \tan(t) \approx 1.60. \]
