What is tan if theta if secant of theta is 7/3?

To find \(\tan(\theta)\) given that \(\sec(\theta) = \frac{7}{3}\), we can use the relationship between secant, tangent, and the Pythagorean identity.

### Step 1: Use the identity
We know that:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
Thus,
\[
\cos(\theta) = \frac{3}{7}
\]

### Step 2: Use the Pythagorean identity
The Pythagorean identity states:
\[
\tan^2(\theta) + 1 = \sec^2(\theta)
\]
Substituting \(\sec(\theta) = \frac{7}{3}\) into the identity:
\[
\tan^2(\theta) + 1 = \left(\frac{7}{3}\right)^2
\]
\[
\tan^2(\theta) + 1 = \frac{49}{9}
\]
\[
\tan^2(\theta) = \frac{49}{9} - 1
\]
\[
\tan^2(\theta) = \frac{49}{9} - \frac{9}{9} = \frac{40}{9}
\]

### Step 3: Solve for \(\tan(\theta)\)
\[
\tan(\theta) = \pm \sqrt{\frac{40}{9}} = \pm \frac{\sqrt{40}}{3} = \pm \frac{2\sqrt{10}}{3}
\]

So, \(\tan(\theta) = \pm \frac{2\sqrt{10}}{3}\).

### Final Answer
\(\tan(\theta) = \pm \frac{2\sqrt{10}}{3}\), depending on the quadrant in which \(\theta\) is located.

Would you like more details on how to determine the correct sign or any other questions?

Here are 5 questions related to the topic:
1. How can you determine the correct sign of \(\tan(\theta)\) based on the quadrant?
2. What are the exact values of \(\sin(\theta)\) and \(\cos(\theta)\) for this problem?
3. How does the value of \(\sec(\theta)\) affect the range of \(\theta\)?
4. What is the significance of the Pythagorean identity in trigonometry?
5. Can \(\sec(\theta)\) ever be less than 1? Why or why not?

**Tip:** Remember that the sign of trigonometric functions depends on the quadrant in which the angle lies.

Learn how to find tan(theta) given sec(theta) = 7/3 with detailed step-by-step instructions. This problem involves trigonometric identities and the Pythagorean identity. Suitable for advanced high school students.

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