Cosecant Study Guide

Cosecant: Cosecant is a trigonometric function that is defined as the reciprocal of the sine function. It is denoted by csc or cosec.

Definition:

The cosecant of an angle θ in a right-angled triangle is calculated as the ratio of the length of the hypotenuse to the length of the side opposite to the angle. In a unit circle, the cosecant of an angle θ is the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

Formula:

The formula for cosecant is:

csc(θ) = 1 / sin(θ)

Graph:

The graph of the cosecant function shows a series of vertical lines where the sine function crosses the x-axis. It has asymptotes at these points, creating a repeating pattern of curves.

Properties:

• The domain of the cosecant function is all real numbers except where sin(θ) = 0, causing division by zero.
• The range of the cosecant function is all real numbers.
• The period of the cosecant function is 2π.

Reciprocal Trigonometric Identities:

Other reciprocal trigonometric identities include:

cot(θ) = 1 / tan(θ)

sec(θ) = 1 / cos(θ)

Examples:

Calculate the value of cosecant for an angle of 30 degrees:

csc(30°) = 1 / sin(30°) = 1 / 0.5 = 2

Calculate the value of cosecant for an angle of 45 degrees:

csc(45°) = 1 / sin(45°) = 1 / √2 ≈ 0.707

Summary:

The cosecant function is the reciprocal of the sine function and is denoted by csc or cosec. It has unique properties and plays a significant role in trigonometry and calculus.