The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Special angles, such as 30°, 45°, and 60° (or $\frac{\pi}{6}$, $\frac{\pi}{4}$, and $\frac{\pi}{3}$ radians), are frequently used in trigonometry. Knowing the exact values of trigonometric functions (sine, cosine, tangent) for these angles is crucial for solving many problems. The unit circle provides a visual representation of these values, where the coordinates of a point on the circle correspond to the cosine and sine of the angle.
This worksheet will test your understanding of these core concepts. Good luck!
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Radian | A. The ratio of the opposite side to the hypotenuse in a right triangle. |
| 2. Unit Circle | B. An angle whose vertex is at the center of a circle. |
| 3. Sine | C. The distance around a circle equal to the radius. |
| 4. Cosine | D. A circle with a radius of 1, centered at the origin. |
| 5. Central Angle | E. The ratio of the adjacent side to the hypotenuse in a right triangle. |
Complete the following paragraph using the words provided: cosine, sine, unit circle, angle, coordinates.
The _________ is a circle with a radius of 1 centered at the origin. For any _________ $\theta$ on the unit circle, the _________ of the point where the terminal side of the angle intersects the circle are given by $(\cos(\theta), \sin(\theta))$. Therefore, the x-coordinate is the _________ of the angle, and the y-coordinate is the _________ of the angle.
Explain how the unit circle helps in determining the signs (positive or negative) of trigonometric functions in different quadrants. Provide an example.
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