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### Given:
- A circle with center \( O \) and a right triangle \( \triangle OCA \) inscribed in the circle.
- \( \overline{OA} \) is the radius of the circle.
- \( \overline{AC} \) is the tangent to the circle at \( C \).
- \( \overline{OC} \) is perpendicular to \( \overline{AC} \).
- The angle \( \angle OAC = \theta \) (let's refer to it as \( \angle A \)).

### Filling in the Blanks:

1. **\( \cos \angle A \) (or \( \cos \theta \))**:
   \[
   \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{OA}{OA} = 1
   \]
   
2. **\( \sin \angle A \) (or \( \sin \theta \))**:
   \[
   \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{OC}{OA} = \frac{OC}{OA}
   \]
   Since \( OC = \text{radius} = OA \times \sin \theta \), but in the unit circle case \( \sin \theta = y \) where \( y \) is the y-coordinate. For a 45-degree triangle, this value could be \( \sin 45^\circ = \frac{1}{\sqrt{2}} \).

3. **For \( \triangle OAC \), \( \tan \angle A \)**:
   \[
   \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{OC}{OA} = \text{Tangent of the angle}
   \]

4. **For \( \triangle ABE \), \( \tan \angle A \)**:
   \( \triangle ABE \) involves another point which is not fully visible, but typically in such diagrams, it represents a line through a point on the circle (perhaps a tangent or secant). Here, you'd calculate the \( \tan \) of the given angle in a similar manner, \( \tan = \frac{\text{opposite}}{\text{adjacent}} \).

5. **The length of the tangent line segment equals the tangent of the**:
   This statement refers to the length of the tangent from a point to the circle. If you have a right triangle \( \triangle OAC \) with the tangent as a side, then \( \text{length of the tangent} = \tan(\text{angle}) \times \text{radius} \).

6. **Pattern of ratios called the**:
   The pattern of ratios around a circle with a radius of 1 refers to the trigonometric functions: sine, cosine, tangent, and their reciprocal functions. This pattern might be referring to the **unit circle** properties or **trigonometric identities**.

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### Relative Questions:
1. How do you calculate \( \sin \) and \( \cos \) for any angle on the unit circle?
2. What is the relationship between the angle \( \theta \) and the length of a tangent from a point outside the circle?
3. What are the properties of a tangent to a circle from a point on the circle?
4. How does the Pythagorean identity relate to the circle in this problem?
5. What is the significance of the unit circle in trigonometry?

### Tip:
In problems involving circles and tangents, it's often useful to remember that the radius to a point of tangency is always perpendicular to the tangent line.

Explore the relationship between trigonometric ratios (sine, cosine, tangent) and tangents in circles. Learn how to calculate these ratios using the unit circle and understand the length of tangents from points outside the circle. Suitable for students in Grades 11-12.

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