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Chapter 6: Problem 67

Find the function value. Round to four decimal places. $$\csc 89.5^{\circ}$$

Short Answer

Expert verified
1.0000

Step by step solution

01

Know the Definition of Cosecant

The cosecant function, \(\text{csc} \( \theta \)\) is the reciprocal of the sine function. That is, \(\text{csc} \( \theta \) = \frac{1}{\text{sin} \( \theta \)}\).
02

Find the Sine of the Angle

Given the angle \(89.5^{\circ}\), use a calculator to find \( \text{sin} \( 89.5^{\circ} \) \). You should get approximately 0.9999619.
03

Calculate the Cosecant Value

Using the definition from Step 1, find \( \text{csc} \( 89.5^{\circ} \) = \frac{1}{\text{sin} \( 89.5^{\circ} \)} \). So, \(\text{csc} \( 89.5^{\circ} \) = \frac{1}{0.9999619} \) which is approximately 1.0000381.

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Key Concepts

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

Reciprocal Trigonometric Functions
Trigonometric functions like \(\text{csc}\theta\) (cosecant) are part of a group called reciprocal trigonometric functions. That means they are defined by taking the reciprocal (or inverse) of another primary trigonometric function.
For instance, the cosecant function is the reciprocal of the sine function. We write this relationship as: \[ \text{csc}\theta = \frac{1}{\text{sin}\theta} \]
It is crucial to understand this for calculating cosecant values correctly. Besides cosecant, there are other reciprocal functions such as:
  • Secant (\text{sec}\theta), which is \(\frac{1}{\text{cos}\theta}\)
  • Cotangent (\text{cot}\theta), which is \(\frac{1}{\text{tan}\theta}\)
Learning these relationships helps simplify the process when working on trigonometry problems.
Calculating Trigonometric Values
To find the value of a trigonometric function, like \(\text{csc} 89.5^{\text{o}}\), a good understanding of angles and their corresponding function values is vital. Here's a systematic approach:
  • First, determine the primary trigonometric function value. For cosecant, you first need the sine value of the given angle.
  • Use a scientific calculator or trigonometric tables to find \(\text{sin} 89.5^{\text{o}}\). You should get approximately 0.9999619.
  • Finally, use the reciprocal relationship to find the desired function value. For \(\text{csc} 89.5^{\text{o}}\), it is \(\frac{1}{\text{sin} 89.5^{\text{o}}}\) which gives about 1.0000381.
This stepwise approach will make it easier to tackle trigonometric problems.
Using a Calculator for Trigonometry
Modern calculators can significantly simplify the process of finding trigonometric values. Here's how to use them effectively:
  • Ensure your calculator is set to the correct mode (degrees or radians) based on the angle type. For this problem, use degrees.
  • Enter the angle value and press the function button (e.g., \(\text{sin}\) for sine).
  • The calculator will display the trigonometric value. For \(\text{sin} 89.5^{\text{o}}\), you get approximately 0.9999619.
  • To find the reciprocal function like cosecant, simply take the inverse of the sine value displayed. Most advanced calculators have a button for this purpose.
    Practice using these calculator functions will make solving trigonometric problems quicker and more accurate.

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Most popular questions from this chapter

Given the function value and the quadrant restriction, find \(\theta\). FUNCTION VALUE = \(\cos \theta=-0.0990\) INTERVAL = \(\left(90^{\circ}, 180^{\circ}\right)\) \(\boldsymbol{\theta}\) = ____

$$\text {Graph each of the following.}$$ $$f(x)=e^{-x / 2} \cos x$$

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$\frac{3}{4} x+\frac{1}{2} y=-5[1.3]$$

Use a calculator in Exercises \(81-84,\) but do not use the trigonometric function keys. Given that \(\sin 65^{\circ}=0.9063\) \(\cos 65^{\circ}=0.4226\) \(\tan 65^{\circ}=2.1445\) find the trigonometric function values for \(115^{\circ} .\)

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=\frac{1}{2} \sec \left(\frac{1}{2} x\right)$$
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