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Chapter 5: Problem 36

Find the approximate value of each expression. Round to four decimal places. $$\csc (1.54)$$

Short Answer

Expert verified
1.0002

Step by step solution

01

Understand the Function

The expression involves the cosecant function, \(\text{csc}\). Recall that \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\).
02

Calculate the Sine

Determine the sine of the angle given. Compute \(\text{sin}(1.54)\) using a calculator or a table of values. \(\text{sin}(1.54) \approx 0.9998\).
03

Compute the Cosecant

Use the sine value to find the cosecant by taking its reciprocal. \(\text{csc}(1.54) = \frac{1}{\text{sin}(1.54)} \approx \frac{1}{0.9998} \approx 1.0002\).
04

Round to Four Decimal Places

Round the result to four decimal places. \(\text{csc}(1.54) \approx 1.0002\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosecant Function
The cosecant function, written as \text{csc}(x), is one of the lesser-known trigonometric functions. It is the reciprocal of the sine function. This means that for any angle x, \text{csc}(x) is defined as \frac{1}{\text{sin}(x)}. This function is particularly useful in various fields, including engineering, physics, and mathematics. Understanding its relationship with the sine function is crucial for solving trigonometric problems. For example, let's find the approximate value of \(\text{csc}(1.54)\).
Sine Function
The sine function, denoted as \(\text{sin}(x)\), is one of the most fundamental trigonometric functions. It represents the y-coordinate of a point on the unit circle as you move an angle x radians from the positive x-axis. To solve our problem of finding \(\text{csc}(1.54)\), we first need to compute \(\text{sin}(1.54)\). Using either a calculator or a table of values, we find that \(\text{sin}(1.54) eq 0.9998\). This will be crucial to our next steps.
Reciprocal Trigonometric Functions
The concept of reciprocal trigonometric functions is essential for understanding functions like the cosecant. In general, if you know one trigonometric function, you can easily find its reciprocal counterpart. For instance, given \(\text{sin}(x)\), you can find \(\text{csc}(x)\) by computing \(\frac{1}{\text{sin}(x)}\). Applying this to \(\text{sin}(1.54)\), which we determined to be 0.9998, we get \(\text{csc}(1.54) = \frac{1}{0.9998} eq 1.0002\).
Approximation Methods
Approximation methods are essential when dealing with trigonometric functions, especially since many trigonometric values are irrational numbers. In our example, we carefully computed \(\text{csc}(1.54)\) and found it to be approximately 1.0002. The key to accuracy in approximation is to round the results correctly. We rounded to four decimal places which can be critical when precision matters. Always ensure your final answer is as close to the true value as possible by using accurate values at each computation step.

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Most popular questions from this chapter

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\csc (x-\pi / 2)$$

Solve each problem. Find \(\sin (\alpha),\) given that \(\cos (\alpha)=3 / 5\) and \(\alpha\) is in quadrant IV.

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=-2 \csc (\pi x-\pi)$$

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=2 \sec x$$

Determine the period and sketch at least one cycle of the graph of each function. $$y=-2 \tan x$$
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