Problem 59 - Precalculus : Building Concepts and Connections

Question

Evaluate the given expressions to four decimal places with a calculator. $$\csc ^{-1}(-3.6)$$

Step-by-Step Solution

Answer

The inverse cosecant of -3.6, rounded off to four decimal places, will be the final answer.

Step 1: Understand the Operation

The expression \( \csc^{-1}(-3.6) \) signifies the operation of finding the inverse cosecant (csc) of the number -3.6.

Step 2: Use a Scientific Calculator

A scientific calculator or mathematical software that can perform this operation is needed. These tools often have inbuilt functions for finding the inverse of trigonometric functions such as cosecant.

Step 3: Perform the Operation

Input the number -3.6 into the function for inverse cosecant and perform the calculation. The tool will provide an output representing the inverse cosecant of -3.6.

Step 4: Report to Four Decimal Places

The result should be rounded off to four decimal places for the final answer.

Key Concepts

Cosecant

The cosecant function, often denoted as \( \csc \theta \), is the reciprocal of the sine function \( \sin \theta \). This means that \( \csc \theta = \frac{1}{\sin \theta} \). In the context of inverse trigonometric functions, \( \csc^{-1}(x) \) refers to the angle \( \theta \) for which the cosecant is \( x \).
For example, if you know that \( \csc \theta = -3.6 \), then \( \theta = \csc^{-1}(-3.6) \), meaning that the sine of this angle would equal the reciprocal of -3.6.
Inverse trigonometric functions, like \( \csc^{-1} \), are used to find angles when you have specific trigonometric values, expanding possibilities in solving geometric and engineering problems. Understanding these functions is crucial for working with angles in different calculations.

Scientific Calculator

A scientific calculator is an essential tool for performing trigonometric calculations. Unlike basic calculators, scientific calculators are equipped with functions to find the sine, cosine, tangent, and their inverses, including those of other trigonometric functions like cosecant.
To evaluate \( \csc^{-1}(-3.6) \), you would:

• Ensure your calculator is in the correct mode, typically radians or degrees, depending on the context.
• Access the inverse trigonometric function menu; the specific steps may vary depending on the calculator model.
• Input the value -3.6 and execute the operation to find the inverse cosecant.

Scientific calculators streamline complex arithmetic and algebraic processes, making them particularly valuable for students and professionals alike.

Decimal Places

Rounding to decimal places is a fundamental skill in mathematics and sciences. It ensures calculations are precise yet manageable. When working with trigonometric values, especially those generated by calculators, results often need rounding to fit specified precision.
For the exercise task of evaluating \( \csc^{-1}(-3.6) \), rounding to four decimal places means:

• Identifying the fourth digit after the decimal point.
• If the digit following it is 5 or more, round up; if it's less, keep it the same.
• For instance, if a result was 2.34567, it would become 2.3457 after rounding.

Working with decimal places helps maintain consistency and accuracy in reporting numerical results, crucial in fields requiring precise measurements.