📈college algebra review

key term - 30°

Definition

30° is an angle that represents one-twelfth of a full circle, or 30 degrees. It is a fundamental angle used in the unit circle, a graphical representation of trigonometric functions.

5 Must Know Facts For Your Next Test

  1. The 30° angle is one of the special angles in the unit circle, along with 0°, 45°, 60°, 90°, and others.
  2. The coordinates of the point on the unit circle at 30° are $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.
  3. The sine of 30° is $\frac{1}{2}$, and the cosine of 30° is $\frac{\sqrt{3}}{2}$.
  4. The tangent of 30° is $\frac{1}{\sqrt{3}}$.
  5. The 30° angle is often used in the study of special right triangles, such as the 30-60-90 triangle.

Review Questions

  • Explain the significance of the 30° angle in the unit circle and how it relates to trigonometric functions.
    • The 30° angle is a fundamental angle in the unit circle, which is a graphical representation of trigonometric functions. The coordinates of the point on the unit circle at 30° are $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, and the trigonometric functions of 30° have specific values: sine is $\frac{1}{2}$, cosine is $\frac{\sqrt{3}}{2}$, and tangent is $\frac{1}{\sqrt{3}}$. These values are important for understanding and working with special right triangles, such as the 30-60-90 triangle, and for visualizing and applying trigonometric concepts.
  • Describe how the 30° angle is used in the study of special right triangles.
    • The 30° angle is a key component of the 30-60-90 triangle, which is a special right triangle with angles of 30°, 60°, and 90°. The side lengths of this triangle have a specific ratio, with the shorter leg being 1 unit, the longer leg being $\sqrt{3}$ units, and the hypotenuse being 2 units. Understanding the properties of the 30-60-90 triangle, including the trigonometric functions associated with the 30° angle, is essential for solving problems involving special right triangles and applying trigonometric concepts in various mathematical and real-world contexts.
  • Analyze the role of the 30° angle in the broader context of the unit circle and its applications in mathematics and beyond.
    • The 30° angle is a fundamental angle in the unit circle, which serves as a powerful tool for visualizing and understanding trigonometric functions. The unit circle provides a geometric interpretation of these functions, allowing for the exploration of their properties and relationships. The 30° angle, along with other special angles, is crucial for this understanding, as it enables the calculation of specific trigonometric values and the application of these concepts in various mathematical and scientific domains, such as physics, engineering, and data analysis. By mastering the properties of the 30° angle within the unit circle, students can develop a deeper appreciation for the elegance and versatility of trigonometry and its far-reaching applications.

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