Problem 222 - Calculus Volume 1

Question

[T] Using \(\mu=2 \sin ^{-1}\left(\frac{1}{M}\right),\) find the Mach number \(M\) for the following angles. $$ \begin{array}{ll}{\text { a. } \quad} & {\mu=\frac{\pi}{6}} \\ {\text { b. }} & {\mu=\frac{2 \pi}{7}} \\ {\text { c. }} & {\mu=\frac{3 \pi}{8}}\end{array} $$

Step-by-Step Solution

Answer

(a) \( M \approx 3.868 \), (b) \( M \approx 2.305 \), (c) \( M \approx 1.800 \).

Step 1: Understand the Formula

The formula given is \( \mu = 2 \sin^{-1}\left(\frac{1}{M}\right) \). We need to solve this equation for \( M \). To do this, we'll take the inverse transformation, \( \sin\left(\frac{\mu}{2}\right) = \frac{1}{M} \), which can be rearranged to solve for \( M \): \( M = \frac{1}{\sin\left(\frac{\mu}{2}\right)} \).

Step 2: Calculate for \(\mu = \frac{\pi}{6}\)

First, evaluate \( \frac{\mu}{2} = \frac{\pi}{12} \). Then find \( \sin\left(\frac{\pi}{12}\right) \). Use a calculator or trigonometric identity to find \( \sin\left(\frac{\pi}{12}\right) = \sin(15°) \approx 0.2588 \). Thus, \( M = \frac{1}{0.2588} \approx 3.868 \).

Step 3: Calculate for \(\mu = \frac{2\pi}{7}\)

Evaluate \( \frac{\mu}{2} = \frac{\pi}{7} \). Find \( \sin\left(\frac{\pi}{7}\right) \). Use a calculator to find \( \sin\left(\frac{\pi}{7}\right) \approx 0.4339 \). Hence, \( M = \frac{1}{0.4339} \approx 2.305 \).

Step 4: Calculate for \(\mu = \frac{3\pi}{8}\)

Evaluate \( \frac{\mu}{2} = \frac{3\pi}{16} \). Find \( \sin\left(\frac{3\pi}{16}\right) \). Using a calculator, \( \sin\left(\frac{3\pi}{16}\right) \approx 0.5556 \). Thus, \( M = \frac{1}{0.5556} \approx 1.800 \).

Key Concepts

Mach number

The Mach number The Mach number is a fascinating concept from the field of fluid dynamics and aerodynamics, named after Austrian physicist Ernst Mach. It is used to describe the speed of an object moving through a fluid or air relative to the speed of sound within that medium. Simply put, the Mach number is the ratio of the object's velocity to the speed of sound. This can be expressed mathematically as:

• Mach number (M) = Speed of the object / Speed of sound

Understanding the Mach number helps you determine whether an object is moving slower than sound (subsonic), at the speed of sound (transonic), faster than sound (supersonic), or much faster than sound (hypersonic). Calculating the Mach number can give critical insights into how an object experiences drag, pressure, and temperature changes while moving at high speeds.

inverse sine function

Understanding the inverse sine function The inverse sine function, denoted as \(ackslash \sin^{-1}(x)\), is one of the essential inverse trigonometric functions. It is crucial for determining the angle whose sine value is a given number. Remember, since the sine function itself is not one-to-one over all real numbers, its inverse only covers a restricted range.

• Range: The range of the inverse sine function is from \( -\frac{\pi}{2}\) to \( \frac{\pi}{2}\), which represents the allowable angles in radians.
• Domain: The domain is from \([-1, 1]\), which aligns with the range of the normal sine function.

When solving problems like the given exercise, using the inverse sine function involves ensuring the angle falls within the valid range, making it easier to solve trigonometric equations by finding the angle \(ackslash \theta\) for a given sine value.

trigonometric identities

Exploring trigonometric identities Trigonometric identities are equations that are always true for any angle. They are incredibly handy when manipulating expressions and solving trigonometric equations. In the exercise, the conversion of the formula from \( \mu = 2\sin^{-1}\left(\frac{1}{M}\right)\) to find \( M \) relies on a few fundamental trigonometric ideas. Some standard trigonometric identities that often prove useful include:

• Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \)
• Double Angle Formulas: like \( \sin(2\theta) = 2\sin\theta\cos\theta \)

For the exercise, understanding \(ackslash \sin\left(\frac{\mu}{2}\right)\) was key. Knowing such identities and principles simplifies the process of rearranging and solving equations, transforming complex trigonometric expressions into something more manageable.

calculator usage

Using calculators for trigonometry While grasping the theory behind trigonometry is vital, leveraging technology can make solving problems quicker and more accurate. Calculators can compute trigonometric functions with high precision, which is especially useful for finding values like \( \sin(\pi/12) \). Here are some calculator tips for trigonometry:

• Radian Mode: Ensure your calculator is set to radian mode if working with angles in radians, as some exercises use radians rather than degrees.
• Inverse Functions: Use the inverse function feature (often \( \sin^{-1}, \cos^{-1}, \tan^{-1} \)) for calculating angles from trigonometric ratios.

For our exercise, the calculated values for \( M \) relied on accurately computing sine values within specified ranges. A couple of button presses can provide exact values, reducing the likelihood of human error.