Question

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 5 \pi / 6} \sin x $$

Short answer

The limit of \(\sin x\) as \(x\) approaches \(5\pi/6\) is \(1/2\).

Step-by-step solution

Identify the function

The function given in the problem is \( \sin x \). It is a trigonometric function. We need to find its limit when \( x \) approaches \( \frac{5\pi}{6} \).

Apply the limit to the function

Applying the limit concept as \( x \) tends to \( \frac{5\pi}{6} \) for \( \sin x \), it means we find the value of \( \sin x \) when \( x = \frac{5\pi}{6} \).

Substitute the value into the function

Replace \( x \) with \( \frac{5\pi}{6} \) in \( \sin x \) to get \( \sin(\frac{5\pi}{6}) \).

Compute the value

Utilizing the known value of the trigonometric function, we find that \( \sin( \frac{5\pi}{6}) \) is \( \frac{1}{2} \).

Key concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It is divided mainly into two areas: Derivatives, which deal with rates of change, and Integrals, which deal with the accumulation of quantities. Within this realm, an indispensable concept is the 'limit', which forms the foundation of calculus. Limits are used to define continuity, derivatives, and integrals. The limit process entails finding the value that a function approaches as the input (or x) approaches some value. In our trigonometric function exercise, we are concerned with finding the limit of the sine function as x approaches \(\frac{5\pi}{6}\).

Trigonometry

Trigonometry is the study of the properties of triangles, especially right-angled triangles. The primary functions in trigonometry are sine (sin), cosine (cos), and tangent (tan), which are all ratios of sides of a right-angled triangle. For any angle, these functions define a value that is crucial for solving geometry and physics problems. In many cases, the exact value of these functions for certain angles are essential parts of the curriculum and are necessary for evaluating the limits effectively.

Limits

In mathematics, a limit captures the behavior of a function as its argument approaches a particular point.
Limits are a fundamental concept in calculus and analysis involving sequences, functions, and series. They are used to define and analyze continuity, derivatives, and integrals. When solving limits in trigonometric functions like in our exercise, knowing precise trigonometric values for specific angles is often the key. Furthermore, understanding different techniques to approach a limit is crucial, particularly when the function behavior is not immediately clear or involves complex or undefined expressions.

Sine Function

The sine function, denoted as sin(x), is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. When finding limits involving the sine function, it's important to understand its periodicity and know that it's bounded between -1 and 1. In our exercise, we are asked to find the limit of sine as x approaches \(\frac{5\pi}{6}\), which requires knowing that the sine of \(\frac{5\pi}{6}\) is an exact value, specifically \(\frac{1}{2}\). This knowledge simplifies the process significantly and doesn't require additional limit laws or techniques.