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Chapter 10: Problem 159

In Exercises \(155-164\), find the exact value or state that it is undefined. $$ \sin \left(2 \operatorname{arccsc}\left(\frac{13}{5}\right)\right) $$

Short Answer

Expert verified
The exact value is \(\frac{120}{169}\).

Step by step solution

01

Understand the Arccosecant

The expression involves \(\operatorname{arccsc}\left(\frac{13}{5}\right)\).This is the angle \(\theta\) such that \(\csc(\theta) = \frac{13}{5}\).Since \(\csc(\theta) = \frac{1}{\sin(\theta)}\), it follows that \(\sin(\theta) = \frac{5}{13}\).
02

Express Sin of Double Angle

We need to find \(\sin(2\theta)\), where \(\theta = \operatorname{arccsc}\left(\frac{13}{5}\right)\).We use the double-angle formula: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\).We have \(\sin(\theta) = \frac{5}{13}\), and we need to find \(\cos(\theta)\).
03

Calculate Cosine of Theta

Use the Pythagorean identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\).Substitute \(\sin(\theta) = \frac{5}{13}\) into the formula:\[\left(\frac{5}{13}\right)^2 + \cos^2(\theta) = 1\] \[\frac{25}{169} + \cos^2(\theta) = 1\] \[\cos^2(\theta) = 1 - \frac{25}{169} = \frac{144}{169}\] Finally, solve for \(\cos(\theta)\):\(\cos(\theta) = \frac{12}{13}\)\ (choosing the positive value since \(csc(\theta) > 0\), indicating \(\theta\) is in the first quadrant).
04

Calculate Sin of Double Angle

Substitute the known values \(\sin(\theta) = \frac{5}{13}\) and \(\cos(\theta) = \frac{12}{13}\) into the double-angle formula:\[\sin(2\theta) = 2 \times \frac{5}{13} \times \frac{12}{13}\]\[\sin(2\theta) = 2 \times \frac{60}{169} = \frac{120}{169}\].Hence, \(\sin(2\theta) = \frac{120}{169}\).

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Key Concepts

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

Arccosecant
The arccosecant (\(\operatorname{arccsc}\)) is the inverse trigonometric function for the cosecant. It's used to find the angle whose cosecant is a given value. In this exercise, \(\operatorname{arccsc}(\frac{13}{5})\) means we are looking for an angle \(\theta\) such that \(\csc(\theta) = \frac{13}{5}\). By definition, \(\csc(\theta) = \frac{1}{\sin(\theta)}\), so \(\sin(\theta)\) must be \(\frac{5}{13}\).
  • Remember: The range for \(\operatorname{arccsc}\) is typically \([\frac{\pi}{2}, \pi]\). However, by convention, angles are often taken in the first quadrant when considering positive values.
  • This ensures that the angle not only satisfies the condition imposed by the cosecant, but also lies within a commonly used interval.
Double-Angle Formula
Trigonometric double-angle formulas allow us to relate single angle trigonometric functions to double angles. Specifically, for sine, the double-angle formula is: \[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]This formula shows that to find the sine of double angles, we first need the sine and cosine values of the original angle.
  • It's important to get both \(\sin(\theta)\) and \(\cos(\theta)\) values correct to plug them into the formula.
  • Use known identities like the Pythagorean Identity to find missing values.
Pythagorean Identity
The Pythagorean Identity is a fundamental relationship in trigonometry that connects sine and cosine:\[\sin^2(\theta) + \cos^2(\theta) = 1\]This identity is derived directly from the Pythagorean Theorem applied to the unit circle. Understanding and using this identity is essential as it helps calculate unknown trigonometric values.
  • In our case, given \(\sin(\theta) = \frac{5}{13}\), we substitute into the identity to solve for \(\cos^2(\theta)\).
  • Once \(\cos^2(\theta)\) is known, we can easily calculate \(\cos(\theta)\).
  • This step is crucial for applying the double-angle formula correctly.
Exact Values
Exact trigonometric values are specific numerical results without needing a calculator to approximate. These are typically either well-known ratios or can be found using identities and formulas.
  • In exercises requiring exact values, you find precise solutions often represented as fractions or square roots.
  • For \(\sin(2\theta)\), once \(\sin(\theta)\) and \(\cos(\theta)\) are determined, substitute these values back into the equation to find the exact result of \(\frac{120}{169}\).
  • Working with exact values ensures clarity and accuracy, especially in theoretical and analytical scenarios.

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Most popular questions from this chapter

In Exercises \(87-106\), find the exact value or state that it is undefined. $$ \operatorname{arccot}\left(\cot \left(\frac{2 \pi}{3}\right)\right) $$

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arccsc}\left(\csc \left(-\frac{\pi}{2}\right)\right) $$

In Exercises \(131-154\), find the exact value or state that it is undefined. $$ \cos (\operatorname{arccot}(3)) $$

In Exercises \(131-154\), find the exact value or state that it is undefined. $$ \cot \left(\arcsin \left(\frac{12}{13}\right)\right) $$

In Exercises \(131-154\), find the exact value or state that it is undefined. $$ \csc \left(\arcsin \left(\frac{3}{5}\right)\right) $$
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