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Chapter 8: Problem 6

Find \(\csc \theta\) if \(\cot \theta=\frac{\sqrt{5}}{2}\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Short Answer

Expert verified
The value of \(\csc \theta\) is \(\frac{3}{2}\).

Step by step solution

01

Understand the Given Values

Recognize that \(\cot \theta\) is given as \(\frac{\sqrt{5}}{2}\) and that \(0 \leq \theta \leq \frac{\pi}{2}\). This means \(\theta\) is in the first quadrant where all trigonometric functions are positive.
02

Express \(\cot \theta\) in Terms of \(\sin \theta\) and \(\cos \theta\)

Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Substituting the given value, we have \(\frac{\cos \theta}{\sin \theta} = \frac{\sqrt{5}}{2}\). So, \(\cos \theta = \frac{\sqrt{5}}{2} \sin \theta\).
03

Use the Pythagorean Identity

The Pythagorean identity states \(\sin^2 \theta + \cos^2 \theta = 1\). Substituting \(\cos \theta = \frac{\sqrt{5}}{2} \sin \theta\), we get \(\sin^2 \theta + \left( \frac{\sqrt{5}}{2} \sin \theta \right)^2 = 1\).
04

Simplify the Equation

Simplify the equation to \(\sin^2 \theta + \frac{5}{4} \sin^2 \theta = 1\). Combine like terms to get \((1 + \frac{5}{4}) \sin^2 \theta = 1\) which simplifies to \(\frac{9}{4} \sin^2 \theta = 1\).
05

Solve for \(\sin \theta\)

Solve for \(\sin \theta\) by isolating \(\sin^2 \theta\), giving \(\sin^2 \theta = \frac{4}{9}\). Thus, \(\sin \theta = \frac{2}{3}\) since \(0 \leq \theta \leq \frac{\pi}{2}\).
06

Find \(\csc \theta\)

Recall that \(\csc \theta = \frac{1}{\sin \theta}\). Substitute \(\sin \theta = \frac{2}{3}\) into the equation, yielding \(\csc \theta = \frac{1}{\frac{2}{3}} = \frac{3}{2}\).

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Key Concepts

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

cotangent
The cotangent function, denoted as \(\romptheta\) or \(\text{cot} \theta\), is one of the six fundamental trigonometric functions. Cotangent is defined as the ratio of the adjacent side to the opposite side in a right triangle. Mathematically, this can be written as:
\(\text{cot} \theta = \frac{\cos \theta}{\sin \theta} \).
Since it is derived from the cosine and sine functions, it's important to understand these relationships fully. In our exercise:
  • We begin with the value of \(\text{cot} \theta = \frac{\root{5}}{2} \).
Substituting this into its trigonometric identity, we need to find how it relates to sine and cosine.
Pythagorean identity
The Pythagorean identity is a fundamental relationship in trigonometry, expressing the relationship between the square of sine and cosine. It is written as:
\(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\).
This identity holds for all angles and is foundational for solving many trigonometric problems.
In our case, we use the given value of \(\text{cot} \theta \theta = \frac{\root{5}}{2} \) and the definition from the first section to substitute \(\text{cos}\theta\). Then the identity becomes:
  • \(\text{sin}^2 \theta + \frac{5}{4}\text{sin}^2 \theta = 1 \)
. Simplifying this, we isolate \(\text{sin}\theta\). This relationship is important because it allows us to discover the value of one trigonometric function when we know another.
cosecant
Cosecant, written as \(\text{csc} \theta \), is the reciprocal of the sine function. It is defined as:
\(\text{csc} \theta = \frac{1}{\text{sin} \theta} \).
This function is useful because it helps to find the length of the hypotenuse relative to the opposite side in a right triangle. When we know \(\text{sin} \theta\), we can easily find \(\text{csc} \theta\) by taking the reciprocal.
  • In our example, once we found \(\text{sin}\theta = \frac{2}{3}\).
  • Therefore, \(\text{csc}\theta = \frac{3}{2}\).
This reciprocal relationship is a general way to switch between sine and cosecant and can be performed straightforwardly with any angle where the sine is known.

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Most popular questions from this chapter

Differentiate the given function. $$f(x)=\tan ^{2}(3 x+1)$$

Differentiate the given function. $$f(x)=e^{-2 x} \cos 3 x$$

Find the indicated integral. $$\int_{0}^{1} x \sin \left(x^{2}\right) d x$$

a. Find the period \(p\), the amplitude \(b\), the horizontal shift \(d\), and the vertical shift \(a\) of the function \(f(x)=5.0+3.0 \cos \left[\frac{\pi}{4}(x-1.5)\right]\) b. Sketch the graph of the function \(f(x)\) in part (a).

Find the indicated integral. $$\int \sin x \cos x d x$$
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