Explore the fundamentals of geometry, focusing on triangles and circles, their properties, and their role in trigonometry. Understand angles in standard position, quadrants of the Cartesian plane, reference angles, and the unit circle. Learn how these concepts are crucial for defining trigonometric functions and their graphs, which are pivotal in mathematics.
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Triangles are polygons with three edges and three vertices, formed by connecting three line segments end-to-end
Trigonometric Functions
Trigonometric functions such as sine, cosine, and tangent explore the relationships between the angles and lengths of triangles
Unit Circle
The unit circle is a fundamental tool in trigonometry that provides a geometric representation of the trigonometric functions
Angles in standard position can be used to define trigonometric functions and understand the relationship between angles and the coordinate plane
Circles are defined by a set of points in a plane that are equidistant from a given point called the center
Trigonometric Functions
Trigonometric functions can also be applied to circles, particularly when examining the unit circle
Reference Angles
Reference angles allow us to find the values of trigonometric functions for any angle by relating it to an acute angle with equivalent trigonometric function values
The unit circle is a powerful tool for visualizing and understanding the trigonometric functions
An angle is formed by two rays with a common endpoint known as the vertex and is measured by the amount of rotation from one ray to the other
Cartesian Coordinate Plane
The Cartesian coordinate plane is divided into four quadrants by the x and y axes
Quadrantal Angles
Quadrantal angles occur at 0, 90, 180, and 270 degrees (or 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) radians) and have special properties in trigonometry
Special Angles
Certain angles, including 0, 30, 45, 60, 90, 180, 270, and 360 degrees (or 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\) radians), are commonly used in trigonometry
Graphical Behavior
The graphs of the sine, cosine, and tangent functions illustrate their periodic and symmetrical nature, with specific intervals for maximum, minimum, or zero values
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Triangle Definition
Polygon with 3 edges, vertices; formed by connecting line segments.
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Trigonometry Focus
Studies relationships of triangle angles, lengths; uses sine, cosine, tangent.
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Circle Characteristics
Set of points equidistant from center; radius is distance to any point on circle.
