Problem 60 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\csc \left(440^{\circ}\right)$$

Step-by-Step Solution

Answer

The value of \(\csc (440^{\circ})\) should be evaluated according to your scientific calculator and rounded to four decimal places.

Step 1: Understand Periodicity of Trigonometric Functions

Realize that trigonometric functions have a period of \(360^{\circ}\). It means that the trigonometric value would be the same after each full rotation which is 360°. Therefore, the angle \(440^{\circ}\) can be rewritten as \(440^{\circ}-360^{\circ}=80^{\circ}\). The \(\csc(440^{\circ})\) is the same as \(\csc(80^{\circ})\).

Step 2: Convert to Sine Function

Understand that cosecant is the reciprocal of the sine function. Hence, this can be written as \(\csc(440^{\circ})= 1/\sin(80^{\circ})\).

Step 3: Evaluate Trigonometric Function with Calculator

Use your scientific calculator to evaluate the sine of 80 degrees. Remember to ensure your calculator is in degrees mode. Then take the reciprocal of the obtained sine value to get the value of the cosecant.

Key Concepts

Periodicity of Functions

Trigonometric functions like sine, cosine, and tangent have a special property known as periodicity. Periodicity means these functions repeat their values in regular intervals. For most trigonometric functions, this interval is a full rotation of a circle or 360 degrees. So, if you have an angle of 440° and want to understand it in terms of periodicity, you subtract 360° to get 80°.
This simplifies the calculation since evaluating trigonometric functions becomes easier at these simplified angles.

• This property is useful in trigonometry as it allows you to evaluate large angles effectively.
• It also enhances understanding of the function's repetitive nature over different intervals.

Embracing periodicity simplifies the evaluation of trigonometric functions at non-standard angles.

Cosecant Function

The cosecant function is one of the lesser-known trigonometric functions. It is essentially the reciprocal of the sine function. This means that the cosecant of an angle is the inverse or "flip" of the sine of that angle. Mathematically, it is expressed as \ \[\csc(\theta) = \frac{1}{\sin(\theta)}\].
Cosecant tells us how a wave repeats over intervals and shows the ratios of lengths in right triangles in various angles.

• It only makes sense to discuss cosecant when the sine value isn’t zero, as division by zero is undefined.
• The function is used in various advanced mathematical analyses, particularly in differential equations and complex number calculations.

Understanding the cosecant function helps unravel other complex trigonometric functions and identities.

Reciprocal Trigonometric Identities

Reciprocal identities involve expressing one trigonometric function in terms of another, specifically as its reciprocal. In the context of trigonometry, these identities help in simplifying and solving equations. For example: \ \[\csc(\theta) = \frac{1}{\sin(\theta)}\], \ \[\sec(\theta) = \frac{1}{\cos(\theta)}\], and \ \[\cot(\theta) = \frac{1}{\tan(\theta)}\].
Mind that using reciprocal identities, you can transform complex expressions into simpler forms, making it easier to compute or manipulate these expressions.

• They are crucial for solving trigonometric equations and proving other identities.
• These identities also underline the interconnectedness between different trigonometric functions.

By leveraging reciprocal identities, you can enhance problem-solving techniques and gain deeper insights into the relationships between trigonometric functions.