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Chapter 6: Problem 10

\(\\{\sin x, \cos x, \tan x\\} \quad\) on \((-\pi / 2, \pi / 2)\)

Short Answer

Expert verified
All real numbers are included in the range of these functions, except at \(x = -\pi / 2\) and \(x = \pi / 2\) where \(\tan x\) is undefined.

Step by step solution

01

Understanding the interval

We start by understanding the given interval \((-\pi / 2, \pi / 2)\). This corresponds to the open interval from negative half of pi to positive half of pi. This falls within the first and fourth quadrants of the unit circle, where all trigonometric functions have real values.
02

Evaluating \(\sin x\)

The \(\sin x\) function is positive in the first and fourth quadrants. Its value begins at 0 when \(x = -\pi / 2\), increases to 1 at \(x = 0\), and then decreases back down to 0 at \(x = \pi / 2\). Thus, the values of \(\sin x\) on the interval are all real numbers between -1 and 1.
03

Evaluating \(\cos x\)

The \(\cos x\) function behaves in a similar manner. It starts out at 0 at \(x = -\pi / 2\), increases up to 1 at \(x = 0\), and then decreases back down to 0 at \(x = \pi / 2\). Therefore, the values of \(\cos x\) on the interval are also all real numbers between -1 and 1.
04

Evaluating \(\tan x\)

The \(\tan x\) function, which is the ratio of \(\sin x\) to \(\cos x\), is undefined at \(x = -\pi / 2\) and \(x = \pi / 2\) so we exclude these points. For the points within the interval, tan(x) takes on all real values. This is because tan(x) increases from -鈭� to 鈭� as x increases from -蟺/2 to 蟺/2.

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Key Concepts

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

Sine Function
The sine function, denoted as \( \sin x \), is one of the primary trigonometric functions. It helps us understand the ratio of the opposite side to the hypotenuse in a right triangle. Let's explore its behavior within the interval \((-\pi/2, \pi/2)\).
  • The sine function starts at 0 when \( x = -\pi/2 \).
  • It increases to 1 as we reach \( x = 0 \), resembling a smooth wave rising.
  • From 0 to \( \pi/2 \), \( \sin x \) gently decreases back to 0.
This pattern illustrates the oscillating nature of the sine function, moving between -1 and 1. Its periodicity and symmetry about the origin make it useful in various applications, from physics to engineering. Importantly, in this interval, \( \sin x \) is always real, providing clarity and predictability to its values.
Cosine Function
The cosine function, represented as \( \cos x \), is closely related to the sine function. It describes the ratio of the adjacent side to the hypotenuse in a right triangle. In the interval of \((-\pi/2, \pi/2)\), cosine shows its unique pattern.
  • The function begins at 0 when \( x = -\pi/2 \).
  • It reaches a peak value of 1 at \( x = 0 \), marking its highest point.
  • Finally, it returns to 0 at \( x = \pi/2 \).
This movement creates an oscillating cycle, like going from a low to a high and back again. Just like the sine, \( \cos x \) in this interval stays within the bounds of -1 and 1. These oscillations help us model phenomena like waves and cycles with precision, and understanding this can be immensely helpful in many scientific fields.
Tangent Function
The tangent function, \( \tan x \), is a bit different from sine and cosine, as it represents the ratio of \( \sin x \) to \( \cos x \). It's important to note the peculiarities of this function in the interval \((-\pi/2, \pi/2)\).
  • Tangent is undefined at \( x = -\pi/2 \) and \( x = \pi/2 \), due to zero values in the denominator, \( \cos x \).
  • Within this interval, \( \tan x \) varies from \(-\infty\) to \(\infty\), sweeping through all real numbers.
Tan's propensity to grow rapidly as it approaches these undefined points requires us to exclude the endpoints. This function is particularly useful in trigonometric identities and calculus, allowing us to model unbounded behaviors such as asymptotes and sharp turns.

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Most popular questions from this chapter

Higher-Order Cauchy-Euler Equations. A differential equation that can be expressed in the form $$ a_{n} x^{n} y^{(n)}(x)+a_{n-1} x^{n-1} y^{(n-1)}(x)+\cdots+a_{0} y(x)=0 $$ where \(a_{n}, a_{n-1}, \ldots, a_{0}\) are constants, is called a homogeneous Cauchy-Euler equation. (The second-order case is discussed in Section \(4.7 .\) ) Use the substitution \(y=x^{r}\) to help determine a fundamental solution set for the following Cauchy-Euler equations: (a) \(x^{3} y^{\prime \prime \prime}+x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=0, x>0\) (b) \(x^{4} y^{(4)}+6 x^{3} y^{\prime \prime \prime}+2 x^{2} y^{\prime \prime}-4 x y^{\prime}+4 y=0, x>0\) (c) \(x^{3} y^{\prime \prime \prime}-2 x^{2} y^{\prime \prime}+13 x y^{\prime}-13 y=0, x>0\) [Hint: \(\begin{aligned} x^{\alpha+i \beta} &=e^{(\alpha+i \beta) \ln x} \\\ &=x^{\alpha}\\{\cos (\beta \ln x)+i \sin (\beta \ln x)\\} . ] \end{aligned}\)

Find a general solution to \(y^{\prime \prime \prime}-3 y^{\prime}-y=0\) by using Newton's method or some other numerical procedure to approximate the roots of the auxiliary equation.
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