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### II. Find the period of the following exercises on the unit circle.
1. \(\cos \left( \frac{17\pi}{4} \right)\)
2. \(\sin \left( \frac{8\pi}{3} \right)\)

For these, we need to reduce the angles to an equivalent angle in the range \([0, 2\pi]\).

1. **Cosine function:**
   \[
   \frac{17\pi}{4} = 4\pi + \frac{\pi}{4}
   \]
   Since cosine is periodic with a period of \(2\pi\), the angle \(4\pi\) can be reduced to 0. So, we evaluate \(\cos \left( \frac{\pi}{4} \right)\).

2. **Sine function:**
   \[
   \frac{8\pi}{3} = 2\pi + \frac{2\pi}{3}
   \]
   Since sine is also periodic with a period of \(2\pi\), we can reduce this to \(\sin \left( \frac{2\pi}{3} \right)\).

### III. Write the definition of the 6 trigonometric functions and identify the sides in the right triangle.

The six trigonometric functions are defined in a right triangle as follows:

- **Sine (\(\sin\))** = opposite / hypotenuse
- **Cosine (\(\cos\))** = adjacent / hypotenuse
- **Tangent (\(\tan\))** = opposite / adjacent
- **Cosecant (\(\csc\))** = hypotenuse / opposite (reciprocal of sine)
- **Secant (\(\sec\))** = hypotenuse / adjacent (reciprocal of cosine)
- **Cotangent (\(\cot\))** = adjacent / opposite (reciprocal of tangent)

### IV. Evaluate the 6 trigonometric functions in the right triangle. Use the Pythagorean theorem to find the missing side.

1. In the given triangle:
   - One leg is 4, and the other is missing.
   - Use the Pythagorean theorem to find the missing hypotenuse, assuming the hypotenuse is \(c\):
     \[
     c = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}
     \]

Now you can calculate all the trigonometric functions for the angle given using these sides.

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Would you like a detailed breakdown of any specific section or further explanation? Here are some questions to explore the topics further:

1. How do we determine the equivalent angle for a given angle outside the unit circle range?
2. What is the significance of the period in trigonometric functions?
3. How are reciprocal trigonometric functions useful in solving triangles?
4. How can the Pythagorean theorem help in solving right triangles?
5. What are some real-world applications of trigonometric functions?

**Tip:** When simplifying angles for sine and cosine, always reduce them within the \([0, 2\pi]\) range to make evaluation easier.

Find the period of trigonometric functions and evaluate the 6 trigonometric functions in a right triangle using given angles and side lengths.

Learn how to find the period of trigonometric functions using the unit circle and evaluate the six trigonometric functions in a right triangle using the Pythagorean theorem. This solution covers key trigonometric concepts and is suitable for high school students in Grades 9-12.

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