What is the significance of the interval [0∘,90∘]?

The interval \([0^\circ, 90^\circ]\) is significant in trigonometry for several reasons:

### 1. **First Quadrant of the Unit Circle:**
   - The interval \([0^\circ, 90^\circ]\) corresponds to the first quadrant of the unit circle. In this quadrant, both the sine and cosine functions are positive, and their values range from 0 to 1 for \(\sin \theta\) and from 1 to 0 for \(\cos \theta\).

### 2. **Monotonic Behavior of Trigonometric Functions:**
   - Within this interval, trigonometric functions like \(\sin \theta\) and \(\cos \theta\) behave monotonically (they either consistently increase or decrease). Specifically, \(\cos \theta\) decreases as \(\theta\) increases from 0° to 90°. This monotonicity ensures that there is only one solution within the interval for equations involving these functions, making calculations straightforward.

### 3. **Practical Applications:**
   - Many practical problems, especially in physics and engineering, involve angles within the first quadrant because it represents typical orientations, directions, or positions. For instance, when considering the angle of elevation or depression, the angle is typically within this range.

### 4. **Trigonometric Identities and Definitions:**
   - For angles within \([0^\circ, 90^\circ]\), trigonometric identities like \(\sec \theta = \frac{1}{\cos \theta}\) hold true without any sign changes. This simplifies problem-solving because you don't need to consider different cases that arise in other quadrants (e.g., negative values).

### 5. **Positive Secant Values:**
   - Since \(\sec \theta = \frac{1}{\cos \theta}\) and \(\cos \theta\) is positive in this interval, \(\sec \theta\) will also be positive. This aligns with the requirement of the given problem, ensuring that the solution found within this interval is valid and meaningful.

Would you like to explore any more details or have further questions? Here are some more topics you might find interesting:

1. How does the behavior of trigonometric functions change in other quadrants?
2. What are some common real-life applications of angles within the first quadrant?
3. Can you explain the relationship between secant and cosine for negative angles?
4. What happens to the trigonometric functions at the boundaries of the interval \(0^\circ\) and \(90^\circ\)?
5. How do you convert an angle from degrees to radians, and why is this useful?

**Tip:** Understanding the properties of trigonometric functions in different quadrants helps in solving complex problems involving angles beyond the first quadrant.

Explore the significance of the interval [0°, 90°] in trigonometry. Understand its importance in the first quadrant of the unit circle, the behavior of trigonometric functions like sine and cosine, and its practical applications in physics and engineering. Learn about trigonometric identities and the positive secant values within this interval.

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