Question

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow \pi / 2} \sin x $$

Short answer

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

Step-by-step solution

Understanding the Nature of Trigonometric Limits

Limits involving trigonometric functions can often be computed by simply using the basic unit circle values of the trigonometric function at the value of the limit. The sine function, denoted as sin(x), gives the length of the side opposite to angle x in a right triangle with hypotenuse 1, also known as an unit circle. Thus, we need to find the value of sin(x) where x is \(\pi/2\).

Finding the Value of Sine Function at \(\pi/2\)

In a unit circle, the angle \(\pi/2\) corresponds to a point at the top of the circle. At this point, the 'height' of the triangle created is 1. This is also the maximum value of the sine function. Therefore, the sine of \(\pi/2\) is 1.

Substituting the Value

Now substitute the value of \(x = \pi/2\) in \(\lim _{x \rightarrow \pi / 2} \sin x\), it becomes: \(\lim_{x \rightarrow \pi / 2} 1 = 1\).

Key concepts

Understanding Trigonometric Limits

When delving into the world of trigonometric limits, it's crucial to grasp the behavior of these functions as they approach specific points. Trigonometric limits are the values that trigonometric functions, such as the sine, cosine, and tangent functions, approach as the input (or angle) gets closer to a particular value. These limits often make use of properties that come from the geometry of the unit circle, and they serve as fundamental building blocks for calculus, especially when dealing with periodic functions.

For instance, the limit of the sine function as the angle approaches \frac{\pi}{2} is tied directly to the unit circle's 'y-coordinate' at that angle. Understanding these relationships is key, as it allows for straightforward calculations of trigonometric limits by reverting to basic trigonometric values. In our exercise example, recognizing that the limit of \(\sin x\) as \(x\) approaches \(\frac{\pi}{2}\) can be found simply by knowing the sine value at that angle is a crucial realization.

The Unit Circle and It’s Significance

The unit circle is a vital concept in trigonometry. It's a circle with a radius of one unit, centered at the origin (0,0) of the coordinate plane. Points on the unit circle represent terminal sides of angles originating from the origin, with their coordinates corresponding to the cosine and sine values of those angles. For example, the point at the top of the unit circle represents a \(90\degree\) or \(\frac{\pi}{2}\) radian angle and has coordinates (0,1), which tells us the cosine of \(\frac{\pi}{2}\) is 0 and the sine is 1.

Therefore, for trigonometric limits that involve angles where the trigonometric function values are well-known based on the unit circle — such as \(0\), \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) — the limits can often be resolved by simply recalling these coordinate pairs. This foundational understanding of how angles correspond to points on the unit circle allows students to swiftly solve for numerous trigonometric limits without delving into more complex calculations.

The Sine Function & Its Maximum Value

The sine function, denoted as \(\sin(x)\), is one of the primary trigonometric functions and represents the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. When this right-angled triangle is inscribed in the unit circle, the hypotenuse is the radius of the circle and the value of the sine corresponds to the 'y-coordinate' of the point on the circle.

In the context of the given exercise, we deal with the sine of \(\frac{\pi}{2}\). This is a special case where the sine function reaches its maximum value, which is 1. This occurs because at an angle of \(\frac{\pi}{2}\), the corresponding point on the unit circle is directly above the origin, at the maximum 'height' possible in the circle - hence the sine value of 1. Understanding that the sine function oscillates between -1 and 1, and knowing where these values occur on the unit circle, is crucial for solving trigonometric limits involving the sine function.

Calculating Limits of Trigonometric Functions

Calculating the limit of a trigonometric function involves determining what value the function approaches as the input angle approaches a specific value. The technique used in our exercise included directly substituting the value of the angle into the sine function, which is possible since we were dealing with a basic unit circle value of \(\sin(\frac{\pi}{2})\).

This substitution method is highly effective when the limit value corresponds to an angle with a well-known sine value. When confronting limits that cannot be resolved by direct substitution, other strategies, such as L'Hôpital's Rule, trigonometric identities, or sandwich theorems, may be required. However, for elementary limits involving trigonometric functions where the angle approaches 0, \(\frac{\pi}{2}\), \(\pi\), or any other standard angle found on the unit circle, the most practical approach often involves recalling sine and cosine values directly from the unit circle. Mastering the calculation of trigonometric limits paves the way for tackling more advanced problems in calculus.