Problem 31 - Calculus

Question

Find a solution to the equation if possible. Give the answer in exact form and in decimal form. $$8=4 \sin (5 x)$$

Step-by-Step Solution

Answer

No real solution exists because the sine function ranges from -1 to 1.

Step 1: Isolate the Sine Function

Start by isolating the sine function on one side of the equation. Given the equation \(8 = 4 \sin(5x)\), divide both sides by 4 to solve for \(\sin(5x)\). \[ \sin(5x) = \frac{8}{4} = 2 \]

Step 2: Evaluate the Feasibility

It's important to note the range of the sine function, which is between -1 and 1 for all angles. Since \(\sin(5x) = 2\), and the sine of any real number cannot be 2, there is no solution for \(x\) in the real numbers.

Key Concepts

Understanding the Sine Function

The sine function is a vital concept in trigonometry. It relates the angle of a right-angled triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. This function is periodic, meaning it repeats its values over regular intervals. The sine function is commonly written as \( \sin(x) \).The sine function can be visualized as a wave that oscillates between -1 and 1. It starts at 0, reaches 1 at \( \frac{\pi}{2} \), descends back to 0 at \( \pi \), reaches -1 at \( \frac{3\pi}{2} \), and completes a full cycle back at 0 when \( x = 2\pi \). This periodicity is crucial when solving trigonometric equations.For any angle \( x \), \( \sin(x) \) can be calculated using:

• Unit circle perspective
• Trigonometric identities
• Graphical representation

This function is instrumental in modeling wave patterns and periodic behavior found in nature and technology.

Range of the Sine Function

One of the fundamental properties of the sine function is its range. The range is the set of all possible values that \( \sin(x) \) can take.Given the unit circle, the sine function produces values ranging from -1 to 1, inclusive. Thus, \( \sin(x) \) never exceeds these boundaries, regardless of the input \( x \). This bounded range is crucial when solving equations like the one in the original exercise. If you find yourself with an equation like \( \sin(x) = 2 \), as encountered in our step-by-step solution, it's clear that such an equation doesn't have a solution in the realm of real numbers. Understanding the range helps quickly determine feasibility, saving time and effort when solving similar problems in the future.

Real Numbers Limitation

Real numbers are a set that includes all the numbers on the number line, encompassing both the rationals, like fractions, and the irrationals, like the square root of 2. Despite covering a vast numerical spectrum, there are still limitations when it comes to function values, like those produced by the sine function.In trigonometric equations, these limitations mean that certain values cannot be achieved. Specifically, with the sine function having a range of [-1, 1], any equation suggesting a sine value outside this range—such as in this exercise where \( \sin(5x) = 2 \)—reveals that no real number solution exists. Recognizing these limitations is vital to efficiently tackle not just trigonometric equations, but any mathematical problem involving specific function properties. When a proposed solution falls outside the established range, it signals an impossibility in real-number terms.