The given image shows a right triangle with a base of 7 units, a height of 8 units, and an unknown hypotenuse. We are tasked with finding the trigonometric values for $\tan \theta$, $\sec \theta$, and $\sin \theta$, where $\theta$ is the angle opposite the side of length 8.

Step 1: Find the Hypotenuse

We can use the Pythagorean theorem to find the hypotenuse $c$, where: $c = \sqrt{7^2 + 8^2} = \sqrt{49 + 64} = \sqrt{113}$

So the hypotenuse $c = \sqrt{113}$.

Step 2: Calculate the Trigonometric Values

1. $\tan \theta$ (opposite over adjacent): $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{7}$

2. $\sec \theta$ (hypotenuse over adjacent): $\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{\sqrt{113}}{7}$

3. $\sin \theta$ (opposite over hypotenuse): $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{\sqrt{113}}$

Final Answer:

• $\tan \theta = \frac{8}{7}$
• $\sec \theta = \frac{\sqrt{113}}{7}$
• $\sin \theta = \frac{8}{\sqrt{113}}$

Would you like more details on any step? Also, here are some follow-up questions for deeper understanding:

1. How can we rationalize the denominator of $\sin \theta$?
2. Can you derive $\cos \theta$ from these values?
3. What are the other trigonometric functions ($\cot \theta$, $\csc \theta$) for this triangle?
4. How would you calculate $\theta$ using an inverse trigonometric function?
5. Why does the Pythagorean theorem apply to this scenario?

Tip: When working with trigonometric functions, always keep in mind which side corresponds to opposite, adjacent, and hypotenuse relative to the angle in question.