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Chapter 5: Problem 41

Find the approximate value of each expression. Round to four decimal places. $$\csc \left(-44.3^{\circ}\right)$$

Short Answer

Expert verified
-1.4352

Step by step solution

01

Understanding the Problem

We need to find the approximate value of \(\text{csc} \left(-44.3^{\circ}\right)\), and round our answer to four decimal places. Cosecant (\(\text{csc}\)) is the reciprocal of sine.
02

Calculate the Sine

First, calculate \(\text{sin}\left(-44.3^{\circ}\right)\). Use a calculator to get the value of this sine function at \(-44.3^{\circ}\).
03

Find the Reciprocal

Since \(\text{csc}\theta = \frac{1}{\sin\theta}\), take the reciprocal of the sine value obtained in Step 2 to find \(\text{csc}\left(-44.3^{\circ}\right)\).
04

Calculating \(\sin(-44.3^{\circ})\)

Using a calculator, \(\sin(-44.3^{\circ}) \approx -0.6965\).
05

Reciprocal Value

To find the cosecant, compute \(\text{csc}\left(-44.3^{\circ}\right) = \frac{1}{\sin(-44.3^{\circ})} = \frac{1}{-0.6965}\).
06

Final Approximation

So, perform the division to get \(\text{csc}\left(-44.3^{\circ}\right) \approx -1.4352\).

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Key Concepts

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

cosecant
The cosecant function, denoted as \(\text{csc}\), is one of the six fundamental trigonometric functions. It is specifically defined as the reciprocal of the sine function. This means for any angle \(\theta\), \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\). Because it is derived from the sine function, it inherits properties from sine, such as its periodicity and the angle's relationship to the unit circle.
When studying the unit circle:
  • \(\text{sin} \theta\) represents the y-coordinate of the point at angle \(\theta\).
  • Consequently, \(\text{csc} \theta\) is determined by how steeply the line from the origin intersects the circle at \(\theta\).
For example, using \(\theta = -44.3^{\text{掳}}\), the corresponding y-coordinate, \(\text{sin}(-44.3^{\text{掳}})\), helps us compute the reciprocal, \(\text{csc}(-44.3^{\text{掳}})\).
reciprocal trigonometric functions
Reciprocal trigonometric functions provide a different perspective on traditional trigonometric ratios by flipping the values.
These functions include:
  • cosecant (\(\text{csc}\)), the reciprocal of sine (\(\text{sin}\)).
  • secant (\(\text{sec}\)), the reciprocal of cosine (\(\text{cos}\)).
  • cotangent (\(\text{cot}\)), the reciprocal of tangent (\(\text{tan}\)).
Reciprocal functions are particularly useful for solving certain types of trigonometric equations and in calculus.
Understanding the relationship between primary trigonometric functions and their reciprocals is essential. For example, by knowing that \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\), we can determine the value of \(\text{csc}(-44.3^{掳})\), as demonstrated in the exercise.
trigonometric calculation
To perform trigonometric calculations, follow a systematic approach. Because trigonometric functions often involve angles, it helps to visualize these on the unit circle.
For example, let's look at calculating \(\text{csc}(-44.3^{掳})\):
  • Start by finding the sine of the angle. Plug \(-44.3^{掳}\) into a calculator to get \(\text{sin}(-44.3^{掳}) \approx -0.6965\).
  • Then, take the reciprocal. This gives \(\text{csc}(-44.3^{掳}) = \frac{1}{\text{sin}(-44.3^{掳})}\). Hence, \(\text{csc} (-44.3^{掳}) \approx \frac{1}{-0.6965} \approx -1.4352\).
Remember to round the final answer to the required number of decimal places (four, in this case).
Using calculators and understanding the foundational steps is key for accurate trigonometric computations.

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Most popular questions from this chapter

A weight hanging on a vertical spring is set in motion with an upward velocity of \(4 \mathrm{cm} / \mathrm{sec}\) from its equilibrium position. Assume that the constant \(\omega\) for this particular spring and weight combination is \(\pi .\) Write the formula that gives the location of the weight in centimeters as a function of the time \(t\) in seconds. Find the period of the function and sketch its graph for \(t\) in the interval \([0,4]\).

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

Find the exact value of each expression for the given value of \(\theta .\) Do not use a calculator. $$\csc ^{2}(2 \theta) \text { if } \theta=\pi / 8$$

Determine the period and range of each function. $$y=2 \sec (x / 2-1)-1$$

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\csc (3 x+\pi)$$
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