How Trigonometric Function Works

The fundamental six trigonometric functions have a domain input value that is an angle of a right triangle and a numeric answer that is the range. 

Trigonometric Functions:

In trigonometry, there are six basic trigonometric functions. Trigonometric ratios are the functions in question. The sine function, cosine function, secant function, co-secant function, tangent function, and co-tangent function are the six fundamental trigonometric functions. The perpendicular side, hypotenuse, and base of a right triangle are employed in trigonometric formulae to determine the sine, cosine, tangent, secant, cosecant, and cotangent values. The domain of the trigonometric function f(x) = sin (commonly known as the ‘trig function’) is the angle θ represented in degrees or radians, and the range is [-1, 1]. All other functions, too, have a domain and range. Calculus, geometry, and algebra all make heavy use of trigonometric functions.

Formulas for Trigonometric Functions

Using the sides of a right-angled triangle, we may get the values of the trig functions using specific formulae. We utilise the shortened version of these functions to create these formulae. Sine is shortened as sin, cosine is abbreviated as cos, tangent is shorthand as tan, secant is abbreviated as sec, and cotangent is abbreviated as cot. The following are the fundamental formulae for finding trigonometric functions:

• sin θ = Perpendicular/Hypotenuse

• cos θ = Base/Hypotenuse

• tan θ = Perpendicular/Base

• sec θ = Hypotenuse/Base

• cosec θ = Hypotenuse/Perpendicular

• cot θ = Base/Perpendicular

Sine and cosecant are reciprocals of each other, as we can see from the formulae above. Cosine and secant, as well as tangent and cotangent, are reciprocal pairings.

Values of Trigonometric Functions:

The domain of trigonometric functions is measured in degrees or radians. In the table below, some of the major values of various trigonometric functions are shown. These standard values of trig functions at specified angles are also known as main values, and they are commonly employed in computations. A unit circle was used to obtain the primary values of trigonometric functions. All trigonometric formulae are satisfied by these numbers.

Four Quadrant Trigonometric Functions:

The angle is an acute angle (90°) and is measured in the anticlockwise direction with reference to the positive x-axis. Furthermore, the distinct quadrants of these trigonometric functions have different numeric signs (+ or -) dependent on the positive or negative axis of the quadrant. In quadrants I and II, the trigonometric functions Sin and Cosec are positive, whereas in quadrants III and IV, they are negative. In the first quadrant, all trigonometric functions have a positive range. Only Quadrants I and III have positive trigonometric functions Tan and Cot, and only Quadrants I and IV have positive trigonometric ratios Cos and Sec.

In the first quadrant, the trigonometric functions have values of θ (90° – θ ). The cofunction identities show how the different complementary trigonometric functions for the angle (90° – θ ) are related.

• sin(90°−θ) = cos θ

• cos(90°−θ) = sin θ

• tan(90°−θ) = cot θ

• cot(90°−θ) = tan θ

• sec(90°−θ) = cosec θ

• cosec(90°−θ) = sec θ

In the second quadrant, the domain value for various trigonometric functions is (π/2 + θ, π – θ), in the third quadrant, it is  (π + θ, 3π/2 – θ), and in the fourth quadrant, it is  (3π/2 + θ, 2π – θ). The trigonometric values for π/2, 3π/2 vary with their complementary ratios, such as

• Sinθ ⇔ Cosθ, 

• Tanθ ⇔ Cotθ, 

• Secθ ⇔ Cosecθ. 

The trigonometric values for π, 2π stay the same.

Trigonometric Functions’ Domain and Range: 

The domain of trigonometric functions is represented by the value of and the range of the trigonometric function is represented by the resultant value. The range is a real numerical value, while the domain values are in degrees or radians. The domain of a trigonometric function is usually a real numerical value, although in some circumstances, a few angle values are eliminated since they result in an infinite range. Periodic functions are trigonometric functions. The domain and range of the six trigonometric functions are shown in the table below.

Identities of Trigonometric Functions

Reciprocal identities, Pythagorean formulae, sum and difference of trig functions identities, formulas for multiple and sub-many angles, and sum and product of identities are all examples of trigonometric functions identities. All of the formulae below can be simply calculated using the right-angled triangle’s side ratio. The fundamental trigonometric function formulas can be used to obtain the higher formulae. Reciprocal identities are commonly utilised to make trigonometric problems easier to understand.

Reciprocal Identities

• cosec θ = 1/sin θ

• sec θ = 1/cos θ

• cot θ = 1/tan θ

• sin θ = 1/cosec θ

• cos θ = 1/sec θ

• tan θ = 1/cot θ

Pythagorean Identities

• Sin2θ + Cos2θ = 1

• 1 + Tan2θ = Sec2θ

• 1 + Cot2θ = Cosec2θ

Conclusion:

In mathematics, trigonometric functions (also known as circle functions, angle functions, or goniometric functions) link the angle of a right-angled triangle to ratios of two side lengths. Navigation, solid mechanics, celestial mechanics, geodesy, and other geodetic disciplines employ them. Because they are among the most basic periodic functions, they are commonly used in Fourier analysis to investigate periodicity.