Sine

Definition

The sine function, denoted as 'sin', is a trigonometric function that represents the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. It is one of the fundamental trigonometric functions, along with cosine and tangent, and is essential in understanding various topics in college algebra.

5 Must Know Facts For Your Next Test

1. The sine function is used to determine the angle of a right triangle given the lengths of the opposite and hypotenuse sides.
2. In the unit circle, the sine function represents the y-coordinate of a point on the circle with a given angle.
3. Sine identities, such as the sum and difference identities, are used to simplify and verify trigonometric expressions.
4. The sine function is periodic, with a period of $2\pi$, meaning it repeats its values every $2\pi$ radians or $360$ degrees.
5. The sine function is essential in solving non-right triangles using the Law of Sines and in parametric equations.

Review Questions

• Explain how the sine function is used in right triangle trigonometry.
  • In right triangle trigonometry, the sine function represents the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. This relationship allows us to determine the angle of a right triangle given the lengths of the opposite and hypotenuse sides, or to find the length of the opposite side given the angle and hypotenuse length. The sine function is one of the fundamental trigonometric ratios, along with cosine and tangent, that are used to solve problems involving right triangles.
• Describe the role of the sine function in the unit circle and how it relates to the x and y-coordinates.
  • In the unit circle, the sine function represents the y-coordinate of a point on the circle with a given angle. The angle is measured in radians, and the sine value corresponds to the vertical distance from the x-axis to the point on the circle. This relationship between the angle, the y-coordinate, and the sine function is essential in understanding trigonometric identities and simplifying trigonometric expressions.
• Analyze how the sine function is used in solving non-right triangles and parametric equations.
  • The sine function is crucial in solving non-right triangles using the Law of Sines, which relates the sides and angles of a triangle. By understanding the properties of the sine function, including its periodicity and the relationships between the trigonometric ratios, students can use the Law of Sines to determine unknown side lengths or angles in non-right triangles. Additionally, the sine function is a key component in parametric equations, where it is used to describe the x and y-coordinates of a point as a function of a parameter, often the angle. This allows for the representation of complex curves and shapes that cannot be easily described using Cartesian coordinates.

Related terms

Cosine:

The cosine function, denoted as 'cos', represents the ratio of the length of the adjacent side to the length of the hypotenuse of a right triangle.

Tangent:

The tangent function, denoted as 'tan', represents the ratio of the length of the opposite side to the length of the adjacent side of a right triangle.

Trigonometric Ratios:

The trigonometric ratios, including sine, cosine, and tangent, are used to describe the relationships between the sides and angles of a right triangle.