Sin θ

Definition

The sine function, denoted as sin θ, is a trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse of a right-angled triangle. It is a fundamental concept in both the study of graphs of the sine and cosine functions, as well as in the understanding of polar coordinate graphs.

5 Must Know Facts For Your Next Test

1. The sine function, sin θ, represents the y-coordinate of a point on the unit circle at the angle θ, measured in radians or degrees.
2. The graph of the sine function is a periodic function, with a period of $2\pi$ radians or $360$ degrees.
3. The amplitude of the sine function is 1, meaning the maximum and minimum values of sin θ are 1 and -1, respectively.
4. The sine function is used to model periodic phenomena, such as the motion of a pendulum or the variation in daylight hours throughout the year.
5. In polar coordinates, the sine function is used to determine the y-coordinate of a point, given the angle and the radius.

Review Questions

• Explain how the sine function, sin θ, is related to the properties of a right-angled triangle.
  • The sine function, sin θ, represents the ratio of the length of the opposite side to the length of the hypotenuse of a right-angled triangle. This means that for a given angle θ in a right-angled triangle, the sine of that angle is equal to the length of the opposite side divided by the length of the hypotenuse. This relationship is a fundamental concept in trigonometry and is used to solve various problems involving right-angled triangles.
• Describe how the sine function, sin θ, is used in the context of graphs of the sine and cosine functions.
  • The sine function, sin θ, is a key component in the graphs of the sine and cosine functions. The sine function produces a periodic wave-like graph, with a period of $2\pi$ radians or $360$ degrees. The amplitude of the sine function is 1, meaning the maximum and minimum values of sin θ are 1 and -1, respectively. The sine function is used to model various periodic phenomena, such as the motion of a pendulum or the variation in daylight hours throughout the year.
• Analyze the role of the sine function, sin θ, in the context of polar coordinate graphs.
  • In polar coordinate graphs, the sine function, sin θ, is used to determine the y-coordinate of a point, given the angle θ and the radius. The x-coordinate of the point is determined by the cosine function, cos θ. Together, the sine and cosine functions are used to convert between rectangular (Cartesian) coordinates and polar coordinates, allowing for the representation of various shapes and curves in the polar coordinate system.

Related terms

Cosine (cos θ): The cosine function, denoted as cos θ, is a trigonometric function that represents the ratio of the length of the adjacent side to the length of the hypotenuse of a right-angled triangle.

Tangent (tan θ): The tangent function, denoted as tan θ, is a trigonometric function that represents the ratio of the length of the opposite side to the length of the adjacent side of a right-angled triangle.

Unit Circle:

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