Cos θ

Definition

The cosine function, denoted as cos θ, is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse of a right triangle. It is a periodic function that oscillates between -1 and 1, and it is widely used in various mathematical and scientific applications, including the study of Graphs of the Sine and Cosine Functions and Polar Coordinates: Graphs.

5 Must Know Facts For Your Next Test

1. The cosine function, cos θ, is an even function, meaning that cos(-θ) = cos(θ).
2. The graph of the cosine function is a sinusoidal curve that oscillates between -1 and 1, with a period of 2π.
3. The cosine function is closely related to the sine function, as they are both fundamental trigonometric functions that can be used to define and describe the properties of right triangles.
4. In polar coordinates, the cosine function is used to determine the x-coordinate of a point on a polar graph, while the sine function is used to determine the y-coordinate.
5. The cosine function is widely used in various fields, such as engineering, physics, and computer science, to model and analyze periodic phenomena, such as waves, oscillations, and rotations.

Review Questions

• Explain the relationship between the cosine function and the properties of right triangles.
  • The cosine function, cos θ, represents the ratio of the adjacent side to the hypotenuse of a right triangle. This means that the value of cos θ is determined by the lengths of the sides of the triangle, with the adjacent side being the reference point. The cosine function can be used to calculate the lengths of the sides of a right triangle, as well as to determine the angle θ, given the lengths of the sides.
• Describe how the cosine function is used in the context of Polar Coordinates: Graphs.
  • In the context of Polar Coordinates: Graphs, the cosine function is used to determine the x-coordinate of a point on a polar graph. The polar coordinate system represents points using an angle θ and a distance r from the origin. The x-coordinate of a point in polar coordinates is given by the formula $x = r\cos\theta$, where r is the distance from the origin and θ is the angle. This relationship between the cosine function and the polar coordinate system allows for the graphing and analysis of various periodic and circular phenomena.
• Analyze how the properties of the cosine function, such as its periodicity and even symmetry, are reflected in the Graphs of the Sine and Cosine Functions.
  • The cosine function, being a periodic function with a period of $2\pi$, is reflected in the graphs of the sine and cosine functions. These graphs exhibit a sinusoidal shape, oscillating between -1 and 1, with the cosine function being shifted horizontally by $\pi/2$ radians (or 90 degrees) relative to the sine function. Additionally, the even symmetry of the cosine function, where $\cos(-\theta) = \cos(\theta)$, is evident in the symmetry of the cosine graph about the y-axis. These properties of the cosine function play a crucial role in the analysis and understanding of the Graphs of the Sine and Cosine Functions.

Related terms

Sine (sin θ): The sine function, denoted as sin θ, represents the ratio of the opposite side to the hypotenuse of a right triangle. It is another fundamental trigonometric function that is closely related to the cosine function.

Tangent (tan θ): The tangent function, denoted as tan θ, represents the ratio of the opposite side to the adjacent side of a right triangle. It is the ratio of the sine function to the cosine function.

Unit Circle:

The unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) on the Cartesian coordinate plane. It is used to define and visualize the trigonometric functions, including the cosine function.