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<<–2/”>a href=”https://exam.pscnotes.com/5653-2/”>h2>CSC: Cosecant

Definition and Relationship to Sine

Cosecant (csc) is one of the six trigonometric functions. It is the reciprocal of the sine function. In a right triangle, the cosecant of an angle is defined as the ratio of the hypotenuse to the side opposite that angle.

Formula:

csc θ = 1 / sin θ

Relationship to Sine:

Angle (θ)	Sine (sin θ)	Cosecant (csc θ)
0°	0	Undefined
30°	1/2	2
45°	√2/2	√2
60°	√3/2	2/√3
90°	1	1

Graph of Cosecant Function:

The graph of the cosecant function is periodic, with a period of 2π. It has vertical asymptotes at every multiple of π. The function is undefined at these points.

Key Features of the Cosecant Graph:

• Period: 2π
• Domain: All real numbers except multiples of π
• Range: (-∞, -1] ∪ [1, ∞)
• Vertical Asymptotes: x = nπ, where n is an integer
• Symmetry: Odd function (symmetric about the origin)

Applications of Cosecant

Cosecant is used in various fields, including:

• Trigonometry: Solving triangles, finding angles and sides
• Physics: Analyzing wave motion, calculating the height of objects
• Engineering: Designing structures, calculating forces
• Navigation: Determining distances and directions

Examples of Cosecant in Action

Example 1: Finding the Cosecant of an Angle

Given a right triangle with an angle of 30° and an opposite side of 5 units, find the cosecant of the angle.

Solution:

1. Find the hypotenuse: Using the sine function, sin 30° = opposite/hypotenuse = 5/hypotenuse. Therefore, the hypotenuse is 10 units.
2. Calculate the cosecant: csc 30° = hypotenuse/opposite = 10/5 = 2.

Example 2: Using Cosecant in Physics

A wave has a wavelength of 10 meters and an amplitude of 2 meters. Find the cosecant of the angle between the wave’s crest and the horizontal.

Solution:

1. Draw a diagram: The wave’s crest forms a right triangle with the horizontal. The amplitude is the opposite side, and the wavelength is the hypotenuse.
2. Calculate the sine: sin θ = opposite/hypotenuse = 2/10 = 1/5.
3. Find the cosecant: csc θ = 1/sin θ = 1/(1/5) = 5.

Frequently Asked Questions (FAQs)

Q1: What is the difference between sine and cosecant?

A: Sine is the ratio of the opposite side to the hypotenuse in a right triangle, while cosecant is the reciprocal of sine, meaning it is the ratio of the hypotenuse to the opposite side.

Q2: When is cosecant undefined?

A: Cosecant is undefined when the sine of the angle is zero. This occurs at multiples of π.

Q3: What is the relationship between cosecant and other trigonometric functions?

A: Cosecant is related to other trigonometric functions through reciprocal identities:

• csc θ = 1 / sin θ
• csc θ = sec (π/2 – θ)
• csc θ = √(1 + cot² θ)

Q4: How do I graph the cosecant function?

A: To graph the cosecant function, start by graphing the sine function. Then, find the points where the sine function is zero. These points will be the vertical asymptotes of the cosecant function. The cosecant function will be positive where the sine function is positive and negative where the sine function is negative.

Q5: What are some real-world applications of cosecant?

A: Cosecant is used in various fields, including trigonometry, physics, engineering, and navigation. For example, it can be used to calculate the height of objects, analyze wave motion, and determine distances and directions.

Table 1: Trigonometric Functions and Their Reciprocals

Function	Reciprocal
Sine (sin)	Cosecant (csc)
Cosine (cos)	Secant (sec)
Tangent (tan)	Cotangent (cot)

Table 2: Values of Cosecant for Common Angles

Angle (θ)	Cosecant (csc θ)
0°	Undefined
30°	2
45°	√2
60°	2/√3
90°	1
120°	2/√3
135°	√2
150°	2
180°	Undefined
210°	-2
225°	-√2
240°	-2/√3
270°	-1
300°	-2/√3
315°	-√2
330°	-2
360°	Undefined