Trigonometric Identities and Quadrants

The third page delves into important trigonometric identities and the behavior of sine and cosine in different quadrants of the coordinate plane. This information is crucial for understanding circonferenza goniometrica formule.

Definition: Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables.

The page presents the fundamental trigonometric identity:

Highlight: cos²(x) + sin²(x) = 1

This identity is the basis for many trigonometric calculations and proofs. The page also provides formulas for expressing sine in terms of cosine and vice versa:

Example: cos(a) = √(1 - sin²(a)) sin(a) = √(1 - cos²(a))

The concept of quadrants is introduced, showing how the signs of sine and cosine change in each quadrant of the coordinate plane. This is essential for solving trigonometric equations and understanding the behavior of trigonometric functions.

Vocabulary: Quadrant - one of four regions in the coordinate plane, divided by the x and y axes.

The page concludes with an example calculating sin(15°) using the half-angle formula and the known value of sin(30°), demonstrating practical application of trigonometric identities.

Example: sin(15°) = sin(30°/2) = 0.5/2 = 0.25

This comprehensive guide provides a solid foundation for understanding the circonferenza goniometrica seno e coseno, essential for students studying trigonometry and circular functions.

Unit Circle and Trigonometric Functions

The first page introduces the circonferenza goniometrica, or unit circle, and its key properties. The unit circle is centered at the origin (0,0) with a radius of 1. It demonstrates the relationship between angles and their sine and cosine values.

Definition: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system.

The page illustrates how angles are measured counterclockwise from the positive x-axis. It shows key angles in both degrees and radians, such as 30°, 45°, 60°, 90°, 180°, 270°, and 360°.

Highlight: The sine of an angle is the y-coordinate on the unit circle, while the cosine is the x-coordinate.

The page also includes formulas for converting between degrees and radians:

Example: To convert from degrees to radians: α = (2π * x) / 360 To convert from radians to degrees: α = (360 * x) / (2π)

Lastly, it provides a visual representation of sine, cosine, and tangent in relation to a right-angled triangle within the unit circle.