Theorems on Limits of Trigonometric Functions

We must first learn how to measure angles before we can apply trigonometric functions. Although both radians and degrees can be used, radians are a more natural measurement since they are directly related to the unit circle, which has a radius of one.

Trigonometric Identities

An equation using trigonometric functions that is true for all angles for which the functions are defined is known as trigonometric identities. The identities can be used to solve or simplify equations.

Graphs and Periods of the Trigonometric Functions

The values of the trigonometric functions repeat as we travel around the unit circle, as we have seen. This trend may be seen in the graphs of the functions. Let P=(x,y) be a point on the unit circle, and be the angle that corresponds to it. The values of the trigonometric functions at and at +2 are the same because the angles and +2 relate to the same point P. Trigonometric functions, as a result, are periodic functions. A function’s period is defined as the smallest positive number p for which f(x+p)=f(x) for all x values in the domain of f. The period of the sine, cosine, secant, and cosecant functions is 2. The period of the tangent and cotangent functions is because they repeat on a longer interval.

The Six Basic Trigonometric Functions

These functions help us to get the coordinates of a point on any circle—not just a unit circle—or to find an angle given a location on a circle. They also define the relationship between a triangle’s sides and angles.

Consider the unit circle centered at the origin and a point P=(x,y) on the unit circle to determine the trigonometric functions. Let be an angle whose initial side is parallel to the positive x-axis and whose terminal side is the line segment OP. In this condition, an angle is considered to be in a standard position. The values of the six trigonometric functions may then be defined in terms of the coordinates x and y.

As we all know, there are six trigonometric functions. The limit of each function tending to a point can be estimated based on the function’s continuity when its domain and range are taken into account.

• Sine Function

The function f(x) = sin(x) is a linear function throughout its whole domain, with all real numbers as its domain.

• Cosine Function

The function f(x) = cos(x) is a continuous function throughout its whole domain, with all real numbers as its domain.

• Tangent Function

The function f(x) = tan(x) is defined for all real numbers except those where cos(x) is equal to 0, that is, for all integers n, the values /2 + n. As a result, it has a domain that includes all real numbers except /2 + n, n € Z.

• Cosec Function

The function f(x) = cosec(x) is determined for all real numbers except for those where sin(x) equals 0, that is, for all integers n. As a result, it has a domain that includes all real numbers except n, n € Z.

The range of this function is (-∞,-1] U [1,+∞).

So, if the cosine function’s limit is computed in its domain, it is always specified and sits between its range.

• Secant Function

The function f(x) = sec(x) is determined for all real numbers except for those where cos(x) is equal to 0, that is, for all integers n, the values /2 + n. As a result, it has a domain that includes all real numbers except /2 + n, n € Z.

This function’s range is (-, -1] U [1, +)

As a result, when the limit of a sec function is calculated in its domain, it is always defined and falls inside its range.

• The Squeeze Theorem

We can evaluate the limits of very simple trigonometric functions using the tools we’ve learned so far, but we can’t yet assess the limits of algebraic functions. The squeeze theorem, the next theorem, is extremely useful for determining basic trigonometric limits. This theorem enables us to compute limits by “squeezing” a function with an unknown limit at a point between two functions with a known limit at that point.

Cot Function

The function f(x) = cot(x) is derived for all real numbers except for those where tan(x) equals 0, that is, for all integers n. As a result, it has a domain that includes all real numbers except n, n € Z.

(-, +) is the range of this function.

When the cotangent function’s limit is determined in its domain, it is always defined and lies within its range.

Key Concepts

• Radian measure is defined as the angle associated with a length 1 arc on the unit circle having a radian measure of 1. The radian measure of an angle having a degree measure of 180 ° is rad.
• The trigonometric functions’ values for acute angles are defined as ratios of two sides of a right triangle in which one of the acute angles is.
• For a generic angle, define (x,y) as a point on a circle with radius r that corresponds to this angle. The trigonometric functions can be expressed as trigonometric ratios involving the variables x, y, and r.
• Trigonometric functions have a regular pattern. Period 2 is the period of the sine, cosine, secant, and cosecant functions. The period of the tangent and cotangent functions is Squeeze’s theorem

Conclusion

We’ve gone through the limitations of trigonometric functions, trigonometric identities as well as other topics.