Problem 25 - APEX Calculus

Question

Evaluate the given limit. $$ \lim _{x \rightarrow \pi / 6} \csc x $$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Understand the Cosecant Function

The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. Thus, \( \csc x = \frac{1}{\sin x} \). We are tasked with finding the limit as \( x \) approaches \( \frac{\pi}{6} \).

Step 2: Substitute into the Limit Expression

To find the limit \( \lim_{x \to \pi/6} \csc x \), substitute the identity for cosecant into the expression to get \( \lim_{x \to \pi/6} \frac{1}{\sin x} \).

Step 3: Evaluate the Sine at \(x = \frac{\pi}{6}\)

The sine of \( \frac{\pi}{6} \) is a known value: \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Step 4: Calculate the Limit Value

Substitute \( \sin \frac{\pi}{6} = \frac{1}{2} \) into the expression \( \frac{1}{\sin x} \) to find \( \lim_{x \to \pi/6} \frac{1}{\sin x} = \frac{1}{\frac{1}{2}} = 2 \).

Step 5: Conclusion

Since \( \sin \frac{\pi}{6} eq 0 \), the limit exists and its value is 2.

Key Concepts

Cosecant Function

The cosecant function is a fundamental concept in trigonometry. It's symbolized as \( \csc x \) and defined as the reciprocal of the sine function. In mathematical terms, this means:

• \( \csc x = \frac{1}{\sin x} \)

Cosecant is especially useful in calculus and trigonometry when dealing with right triangles and waves. Since the sine function represents the ratio of the opposite side over the hypotenuse in a right triangle, cosecant would be the hypotenuse over the opposite side.

Understanding \( \csc x \) helps when solving problems that involve trigonometric expressions. It provides insight into the behavior of functions where sine values become pivotal.

Trigonometric Limits

Limits involving trigonometric functions serve as the foundation for understanding calculus applications in physics and engineering. More specifically, a trigonometric limit seeks to find the behavior of a trigonometric function as the variable approaches a particular value.

• This is often to determine continuity or find the exact value of a function at a boundary point.
• In more advanced calculus, it helps evaluate definite and indefinite integrals and derivatives involving trigonometric functions.

When calculating trigonometric limits, we use standard limit evaluation techniques, such as substitution or known values of trigonometric functions. For the limit \( \lim _{x \rightarrow \pi / 6} \csc x \), knowing \( \sin \frac{\pi}{6} = \frac{1}{2} \) helps directly calculate the limit value.

Reciprocal Trigonometric Functions

Trigonometric functions have reciprocals that open up more possibilities in mathematical computations and solutions. By using reciprocals:

• Cosecant (\( \csc x \)) is the reciprocal of sine.
• Secant (\( \sec x \)) is the reciprocal of cosine.
• Cotangent (\( \cot x \)) is the reciprocal of tangent.

These functions allow us to convert certain trigonometric expressions into forms that might be easier to deal with. For solving limits, integrating, or differentiating, knowing the reciprocals can be an incredible simplification.

In the context of limits, if you can express a trigonometric function in terms of its reciprocal, like \( \csc x \) as \( \frac{1}{\sin x} \), you can make benefit from known values of \( \sin x \) and manipulate the expression so it becomes more manageable.