sketch each angle in standard position draw an arrow representing the correct amount of of rotation l. Find the measure of two other angles one positive and one negative that are coternminal with given angle give the quadrant of each angle 89

To solve the problem involving the angle of \(89^\circ\):

### 1. **Sketching the Angle in Standard Position:**
- **Standard Position:** The angle is drawn starting from the positive x-axis, with the vertex at the origin.
- **Angle of \(89^\circ\):** 
  - This angle is almost a right angle, just 1 degree short of 90 degrees.
  - **Quadrant:** The terminal side of the angle \(89^\circ\) lies in the **first quadrant**.
  - **Rotation Direction:** The arrow representing the rotation will rotate counterclockwise from the positive x-axis, stopping just before the positive y-axis.

### 2. **Finding Coterminal Angles:**
Coterminal angles can be found by adding or subtracting multiples of \(360^\circ\) to the given angle.

- **Positive Coterminal Angle:**
  \[
  89^\circ + 360^\circ = 449^\circ
  \]
  - This angle is coterminal with \(89^\circ\) but represents a full rotation plus an additional \(89^\circ\), so it also lies in the **first quadrant**.

- **Negative Coterminal Angle:**
  \[
  89^\circ - 360^\circ = -271^\circ
  \]
  - This angle represents a clockwise rotation of \(271^\circ\), which places it in the **third quadrant**.

### 3. **Summary of the Quadrants:**
- **\(89^\circ\):** First Quadrant.
- **\(449^\circ\):** First Quadrant (coterminal angle).
- **\(-271^\circ\):** Third Quadrant (coterminal angle).

Would you like to see a sketch of these angles or need further details?

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Here are 5 related questions to expand on this topic:

1. What is the general formula to find all coterminal angles for a given angle?
2. How do you determine the quadrant for any given angle in standard position?
3. What are the characteristics of angles in the second quadrant?
4. How do you convert an angle in degrees to radians, and why is this conversion important?
5. What are reference angles, and how are they used in trigonometry?

**Tip:** Coterminal angles are useful in trigonometry for simplifying problems, especially when dealing with periodic functions like sine and cosine.

Explore how to sketch angles in standard position, find coterminal angles, and determine their quadrants with a detailed explanation for an angle of 89 degrees. Suitable for high school students learning trigonometry and angle properties.

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