Problem 57 - Precalculus with Calculus Previews

Question

Explain why there are no real numbers \(x\) satisfying the equation \(9 \csc x=1\).

Step-by-Step Solution

Answer

There are no real solutions because the sine function's range is [-1, 1], which does not include 9.

Step 1: Convert to Sine Function

The equation is given as \(9 \csc x = 1\). We know \(\csc x\) is the reciprocal of \(\sin x\), so \(\csc x = \frac{1}{\sin x}\). Substitute this in the equation: \(9 \times \frac{1}{\sin x} = 1\).

Step 2: Simplify the Equation

Simplify the equation from Step 1: \(\frac{9}{\sin x} = 1\). To clear the fraction, multiply both sides by \(\sin x\) to get \(9 = \sin x\).

Step 3: Analyze the Sine Function Values

The sine function \(\sin x\) has a range of values between -1 and 1, inclusive. Since \(9\) is outside of this range, \(\sin x = 9\) is not possible for a real number \(x\).

Step 4: Conclusion

Since the value \(9\) is not in the range of the sine function, there are no real solutions \(x\) that satisfy the equation \(9 \csc x = 1\).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is one of the six primary trigonometric functions. It is defined as the reciprocal of the sine function. Therefore, \( \csc x = \frac{1}{\sin x} \). Recall that for all angles \( x \) where \( \sin x eq 0 \), the cosecant function is defined. This function is particularly useful because it helps in solving equations and understanding relationships in trigonometry.When dealing with reciprocal values like the cosecant, it’s important to remember that the cosecant function will not exist when the sine function equals zero. This is because dividing by zero is undefined. For example:

• If \( \sin x = 0 \), \( \csc x \) is undefined.
• Common angles where \( \csc x \) is defined are multiples of \( \frac{\pi}{2} \).

This means that understanding the domains where these functions are defined is crucial when solving trigonometric equations. The equation in our exercise, after converting from \( \csc x \) to \( \sin x \), leads to an impossibility because it asks for a sine value outside its possible range.

Sine Function

The sine function, represented as \( \sin x \), is a fundamental concept in trigonometry. This function is about finding the y-coordinate of a point on the unit circle corresponding to an angle \( x \).Sine functions are periodic and their values repeat every \( 2\pi \) radians. The sine curve oscillates smoothly between -1 and 1. This oscillation results from its definition based on circular movement.When we explore equations involving the sine function, we often need to remember:

• The sine of any angle is computed as: \( \sin x = \frac{\text{opposite}}{\text{hypotenuse}} \) in a right triangle.
• The range of the sine function is from -1 to 1.
• For instance, \( \sin 0 = 0 \) and \( \sin \frac{\pi}{2} = 1 \).

In our context, identifying that \( \sin x = 9 \) is impossible highlights the importance of understanding the sine function's natural range. Such knowledge enables us to immediately recognize when an equation has no real solutions, as is the case here.

Range of Trigonometric Functions

The range of trigonometric functions refers to the set of possible output values these functions can take. This concept is key to understanding why certain equations, like \( 9 \csc x = 1 \), have no solutions.Focusing on the sine function, its range is limited to values between -1 and 1, inclusive. This means:

• Any real number input to the sine function will result in an output somewhere within this range.
• If an equation suggests a sine output outside this range, it signals no real \( x \) can satisfy it.

The other trigonometric functions, like the cosine, tangent, and cosecant, each have their distinct ranges. For example, while tangent can theoretically take any real number, its reciprocal, the cotangent, is undefined at certain points. Thus, knowing ranges allows students to judge equations quickly, spotting whether solutions make any sense given the inherent limits of the functions involved.