Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 51-52: Solve the equation for \(x\) in terms of \(y\) if \(0

Step-by-Step Solution

Answer

\( x = \sin^{-1}\left( \frac{4}{7} \sin y \right) \)."

Step 1: Cross Multiply the Equation

The given equation is \( \frac{4}{\sin x} = \frac{7}{\sin y} \). To eliminate the fractions, we cross-multiply to get: \( 4 \sin y = 7 \sin x \).

Step 2: Solve for \( \sin x \)

Rearrange the equation from Step 1: \( \sin x = \frac{4}{7} \sin y \). This expresses \( \sin x \) in terms of \( \sin y \).

Step 3: Use the Inverse Sine Function

Since \( x \) is in the range \( 0 < x < \pi \), take the inverse sine of both sides: \( x = \sin^{-1}\left( \frac{4}{7} \sin y \right) \). This step gives \( x \) in terms of \( y \).

Key Concepts

Understanding Inverse Sine

The inverse sine function often plays a pivotal role in solving trigonometric equations. Typically denoted as \( \sin^{-1} \), this function helps to find an angle whose sine is a given value. It's crucial when moving from a sine value back to an angle, especially in mathematical problems where an angle needs to be determined from its sine.Here are some important aspects of inverse sine:

• Definition: \( \sin^{-1}(x) \) gives the angle \( \theta \) such that \( \sin(\theta) = x \).
• Range: The output of \( \sin^{-1}(x) \) is limited to \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) . This means it only outputs angles within the first and fourth quadrants of the unit circle.
• Application: In the given problem, after transforming \( \sin x \) using cross-multiplication, \( x \) is isolated using \( \sin^{-1} \), which allows us to express \( x \) in terms of \( y \).

This tool is particularly useful in real-world applications like physics and engineering where periodic and circular motion is measured.

Grasping Cross-Multiplication

Cross-multiplication is a key algebraic technique used to solve equations that involve fractions. The basic principle involves eliminating fractions by multiplying them across each other, effectively simplifying the given equation.The steps for performing cross-multiplication are straightforward:

• Step 1: Identify the equation in the form \( \frac{a}{b} = \frac{c}{d} \).
• Step 2: Cross-multiply to yield \( a \times d = b \times c \).
• Application: In our problem, starting with \( \frac{4}{\sin x} = \frac{7}{\sin y} \), cross-multiplying gives us \( 4 \sin y = 7 \sin x \). This eliminates the fractions and helps us focus on the core calculations.

Cross-multiplication simplifies complex equations, making them easier to manipulate as seen in our problem. Once free of fractions, equations are much more forgiving and straightforward to solve.

Solving for x Clearly Explained

In trigonometric equations, isolating variables is often essential. To solve for \( x \) in our exercise, we follow a logical sequence, ensuring that every step leads us closer to the expression of one variable in terms of another.Here's how you solve for \( x \):

• Start: With the cross-multiplied equation \( 4 \sin y = 7 \sin x \).
• Rearrange: Solve for \( \sin x \) by dividing both sides by 7: \( \sin x = \frac{4}{7}\sin y \).
• Apply Inverse Sine: To isolate \( x \), take the inverse sine of both sides, giving \( x = \sin^{-1}\left(\frac{4}{7}\sin y\right) \).
• Complete: Now, \( x \) is expressed in terms of \( y \), ready for any specific values or further calculations that may be necessary.

Solving for \( x \) involves breaking down complex equations into simpler, manageable forms. By rearranging and applying trigonometric identities methodically, obtaining solutions becomes a structured journey, not a daunting task.