Let theta be an acute angle of a right triangle for which tan theta = 4/3 Find sec theta?

To find \(\sec \theta\), given that \(\tan \theta = \frac{4}{3}\), let's follow these steps:

1. **Use the definition of \(\tan \theta\):**
   \[
   \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}
   \]
   This means that in the right triangle, the length of the side opposite \(\theta\) is 4, and the length of the side adjacent to \(\theta\) is 3.

2. **Find the hypotenuse using the Pythagorean theorem:**
   \[
   \text{hypotenuse} = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
   \]

3. **Use the definition of \(\sec \theta\):**
   \[
   \sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{5}{3}
   \]

Thus, \(\sec \theta = \frac{5}{3}\).

Would you like more details or have any questions? Here are some related problems to consider:

1. If \(\sin \theta = \frac{4}{5}\), what is \(\cos \theta\)?
2. Find \(\cot \theta\) given that \(\tan \theta = \frac{4}{3}\).
3. If \(\sec \theta = \frac{5}{3}\), find \(\csc \theta\).
4. What is the value of \(\cos^2 \theta + \sin^2 \theta\)?
5. Find the value of \(\theta\) if \(\tan \theta = \frac{4}{3}\).

**Tip:** In any right triangle, the Pythagorean theorem is a powerful tool to relate the sides when trigonometric ratios are given.

Learn how to find sec theta in a right triangle where tan theta equals 4/3. This problem involves using trigonometric ratios and the Pythagorean theorem to determine secant theta. Suitable for high school students studying trigonometry.

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