Trigonometric Functions Of Any Angle

In general, trigonometric functions deal with right-angled triangles, where ratios of the two sides of the triangle are used to determine the trigonometric functions. Trigonometric functions can also be called circular functions or goniometric functions and mainly consist of six types of functions–sine, cosine, tangent, cosecant, secant, and cotangent, of which sine, cosine, and tangent are the three basic trigonometric functions. These functions are the ratios of the sides of the triangle. 

In a right-angled triangle, if an angle is θ, the side opposite to θ is called perpendicular, and the side on which the perpendicular stands is the base. Moreover, the side opposite to the right-angled is called the hypotenuse.

Sine Function

The sine function is the ratio of the side opposite to θ or also known as perpendicular to that of the hypotenuse of that triangle. In short, we can write sine θ as sin θ. The sine function is: 

sin θ = Perpendicular/Hypotenuse      or      

sin θ  = P/h

In a unit circle of radius 1 and with the centre (0,0), it can also be used to define a real number in terms of the sine function.

We can understand this by the following figure , where the sine of angle θ is equal to the y-coordinate of the unit circle. 

Unwrapping this unit circle, we get a sinusoidal graph, which is a periodic graph. This means that the sinusoidal wave will repeat itself after every 2π period. This 2π period is divided into four parts–the first part is from 0 to π/2, which is the first quadrant; the second quadrant is from π/2 to π; the third quadrant is from π to 3π/2; and the last quadrant is from 3π/2 to 2π, as seen below. The sine function is positive in the first two quadrants and negative in the third and fourth quadrants.

 In the sine graph above, we can see that the domain of the function is all the real numbers which are denoted by the x-axis, but the range is from [-1,1], which is denoted by the y-axis in the graph. This means that the value of the sine function lies between [-1,1], i.e., -1 ≤ sin x ≤ 1. 

Cosine Function

The cosine function is the ratio of the base of the triangle to that of the hypotenuse of that triangle. In short, we can write cosine θ as cos θ. 

The cosine function is: 

cos θ = Base/Hypotenuse      or  cos θ  = b/h

A unit circle of radius 1 and with centre (0,0) can also be used to define a real number in terms of the cosine function,  where the cosine of angle θ is equal to the x-coordinate of the unit circle. 

Unwrapping this unit circle, we get an up-down graph, which is a periodic graph. This means that the sinusoidal wave will repeat itself after every 2π period. This 2π period is divided into four parts: the first is from 0 to π/2, which is the first quadrant, the second quadrant is from π/2 to π, the third quadrant is from π to 3π/2, and the last quadrant is from  3π/2 to 2π, as seen below. The cosine function is positive in the first and last quadrant and negative in the second and third quadrant.

 In the cosine graph above, we can see that the domain of the function is all the real numbers, which is denoted by the x-axis, but the range is from [-1,1], which is denoted by the y-axis. This means that the value of the cosine function lies between [-1,1], i.e., -1 ≤ cos x ≤ 1.

Tangent Function

The tangent function is one of the basic functions of trigonometry. It is the ratio of the perpendicular to the base of the right-angled triangle. In other words, it is the ratio of the sine function to the cosine function. The tangent θ, in short, can be written as tanθ. 

tan θ = (Perpendicular )/Base   =  p/b  or  tan θ = (sin θ)/(cos θ)

To better understand the tangent function, we can refer to the unit circle of 1 unit radius with centre (0,0) in the figure below:

 The tangent function is also a periodic function that repeats itself after every π period. The range of the tangent function is all the real numbers, i.e., (-∞,∞), and the domain of the tangent function is all the real numbers except odd multiples of  π/2, i.e., (2n+1)π/2. From the graph below, we can observe that the tan function is positive only in the first and third quadrant and gives negative values in the second and fourth quadrant.

Cosecant Function

The cosecant function is the reciprocal of the cosine function, meaning it is the ratio of the hypotenuse of the triangle to that of the base of that triangle. In short, we can write cosecant θ as cosec θ. The cosecant function is: 

cosecant θ = Hypotenuse/Perpendicular  = h/b or cosec θ = 1/sinθ

Its domain is all the real numbers except nπ and the range is (-∞,-1]∪[1,∞). It is positive in the first and second quadrants and gives negative values in the third and fourth quadrants.

Secant Function

The cosecant function is the reciprocal of the cosine function, meaning it is the ratio of the hypotenuse of the triangle to that of the base of that triangle. In short, we can write the cosecant θ as cosec θ. The cosecant function is: 

secant θ = Hypotenuse/Base  = h/b or sec θ = 1/cosθ

Its domain is all the real numbers except (2n+1)π/2, and the range is (-∞,-1]∪[1,∞). It is positive in the first and fourth quadrants and gives negative values in the second and third quadrants.

Cotangent Function

The cotangent function is one of the basic functions of trigonometry. It is the ratio of the base to the perpendicular of the right-angle triangle. In other words, it is the ratio of the cosine function to the sine function. Cotangent θ, in short, can be written as tan θ. 

Cotangent  θ = (Base )/Perpendicular   =  b/p               or  cot θ=   cosθ/(sin θ)

Its domain is all the real numbers except nπ, and the range is all the real numbers. It is positive in the first and third quadrants and gives negative values in the second and fourth quadrants.

Conclusion

Trigonometric functions are periodic functions whose values repeat themselves after a fixed period. Trigonometric functions are defined for a fixed domain and range, as previously discussed. The point at which the function approaches infinity is the point where the function will not be defined.