Given that $r$ represents an angle such that $0^\circ < r < 90^\circ$, this tells us that:

1. $r$ is a positive angle.
2. The angle $r$ is an acute angle because it is less than $90^\circ$.

In trigonometry, angles within this range have some important properties:

• Sine and Cosine: For $0^\circ < r < 90^\circ$, both $\sin(r)$ and $\cos(r)$ are positive.
• Tangent and Cotangent: The tangent function, $\tan(r) = \frac{\sin(r)}{\cos(r)}$, is also positive in this range.
• Reciprocal Functions: The reciprocal functions, cosecant $\csc(r) = \frac{1}{\sin(r)}$, secant $\sec(r) = \frac{1}{\cos(r)}$, and cotangent $\cot(r) = \frac{1}{\tan(r)}$, are all positive as well.

Would you like to explore more about trigonometric properties of acute angles or have any specific questions?

Relative Questions to Consider:

1. What are the values of sine, cosine, and tangent for specific angles like $30^\circ$, $45^\circ$, and $60^\circ$?
2. How do the graphs of trigonometric functions look for angles between $0^\circ$ and $90^\circ$?
3. Why are all trigonometric ratios positive in the first quadrant?
4. How are acute angles used in solving right triangle problems?
5. What are some real-life applications where acute angles are important?

Tip:

Remember that knowing the properties of angles in different quadrants is crucial for solving trigonometric equations and understanding their behavior across the unit circle.