Trigonometric functions are transcendental functions in which the mathematical expression solely consists of a trigonometric operation such as sine, cosine, tangent, etc.
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The trigonometric functions are transcendental functions that deal with the relationship between a right triangle's interior angle and sides. These are also called circular functions. Let's discuss these further:

Reviewing The Right Triangle

Six trigonometric functions exist sine, cosine, tangent, cosecant, secant, and cotangent. We define these as the link between one of the non-90-degree-interior angles with the ratios of the sides of a right triangle.

If we want to refresh our memory on this topic, let's discuss it here.

Trigonometric Functions

Because a trigonometric operation takes an angle as an input and spits out a ratio as an output, these operations are mathematical functions. Let's discuss it based on its three viewpoints:

Analytic Perspective

In algebraic terms, we can express the general form of the six trigonometric functions as:

\(f(x)=a\sin{(b\theta+c)}+d\)

\(f(x)=a\cos{(b\theta+c)}+d\)

\(f(x)=a\tan{(b\theta+c)}+d\)

\(f(x)=a\csc{(b\theta+c)}+d\)

\(f(x)=a\sec{(b\theta+c)}+d\)

\(f(x)=a\cot{(b\theta+c)}+d\)

  • \(a\), \(b\), \(c\), and \(d\) are constants that modifies the function
  • \(\theta\) is the interior angle of the right triangle
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Numerical Perspective

Say we create a table of values of the different trigonometric functions. Let the inputs be angles (in degrees or radians), and the output would be the ratios. We can find these values using a scientific calculator or a trigonometric table:

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One key observation of trigonometric functions is that it is cyclic. The output for 0 degrees is the same when the angle input is 360 degrees, then 720 degrees, and so forth.

Graphical Perspective

We can further see the cyclic pattern of trigonometric functions using graphs. Assuming we have the trigonometric function in its basic form, below are vital observations among the plots:

Sine

Sine Function

  • \(f(x)=\sin(\theta)\)
  • Zero and 180 degrees (0 radians and \(\pi\) radians) will have a ratio value of 0
  • 90 degrees (\(\frac{\pi}{2}\) radians) will correspond to a ratio value of 1
  • 270 degrees (\(\frac{3\pi}{2}\) radians) will have a ratio value of -1

Cosine

Cosine Function

  • \(f(x)=\cos(\theta)\)
  • Zero degrees (0 radians) will have a ratio value of 1
  • Ninety and 270 degrees (\(\frac{\pi}{2}\) radians and \(\frac{3\pi}{2}\) radians) will correspond to a ratio value of 0
  • 180 degrees (π radians) will have a ratio value of -1

Tangent

Tangent Function

  • \(f(x)=\tan(\theta)\)
  • Zero and 180 degrees (0 radians and π radians) will have a ratio value of 0.
  • Ninety and 270 degrees (\(\frac{\pi}{2}\) radians and \(\frac{3\pi}{2}\) radians) are undefined.

Cosecant

Cosecant Function

  • \(f(x)=\csc(\theta)\)
  • Zero and 180 degrees (0 radians and π radians) are undefined.
  • 90 degrees (\(\frac{\pi}{2}\) radians) will have a ratio value of 1
  • 270 (\(\frac{3\pi}{2}\) radians) degrees will have a ratio value of -1

Secant

Secant Function

  • \(f(x)=\sec(\theta)\)
  • Zero degrees (0 radians) will have a ratio value of 1
  • Ninety and 270 degrees (\(\frac{\pi}{2}\) radians and \(\frac{3\pi}{2}\) radians) are undefined.
  • 180 (\(\frac{\pi}{2}\) radians) degrees will have a ratio value of -1

Cotangent

Cotangent Function

  • \(f(x)=\cot(\theta)\)
  • Zero and 180 degrees (0 radians and π radians) are undefined.
  • Ninety and 270 degrees (\(\frac{\pi}{2}\) radians and \(\frac{3\pi}{2}\) radians) will have a ratio value of 0.

Summary

The trigonometric functions are transcendental functions that deal with the relationship between a right triangle's interior angle and sides.
Because a trigonometric operation takes an angle as an input and spits out a ratio as an output, these operations are mathematical functions. 
In algebraic terms, we can express the general form of the six trigonometric functions as (1) \(f(x)=a\sin{(b\theta+c)}+d\), (2) \(f(x)=a\cos{(b\theta+c)}+d\), (3) \(f(x)=a\tan{(b\theta+c)}+d\), (4) \(f(x)=a\csc{(b\theta+c)}+d\), (5) \(f(x)=a\sec{(b\theta+c)}+d\), and (6) \(f(x)=a\cot{(b\theta+c)}+d\)
One key observation of trigonometric functions is that it is cyclic. The output for 0 degrees is the same when the angle input is 360 degrees, then 720 degrees, and so forth.
We can further see the cyclic pattern of trigonometric functions using graphs.

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1

What are Polynomial Functions?

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Polynomial functions are algebraic functions in which the mathematical expression is a polynomial - terms consisting of coefficients and variables that are added or subtracted together.

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2

Types of Mathematical Functions

Concept

Functions can be classified into certain types. One common classification is to categorize it as either algebraic or transcendental. We'll explore the difference between the two.

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Created On
June 5, 2023
Updated On
February 23, 2024
Contributors
Edgar Christian Dirige
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