Trigonometric functions

Trigonometric functions are functions based on the ratios in a right-angled triangle. The three most well-known trigonometric functions are sine, cosine, and tangent.

 

These functions are periodic, which means that they repeat at fixed intervals. They play a central role in mathematics, physics, and engineering, especially in connection with waves, rotations, and angles.

 

Sine function

The sine function describes the ratio between the opposite side and the hypotenuse in a right-angled triangle.

It is written in this form:

 

$$ \Large y = \sin(x) $$

 

Characteristics:

  • The period is \(2\pi\)
  • The amplitude is 1
  • Starts at \((0,0)\)
  • The curve oscillates between -1 and 1

 

Example: Values for \(\large y = \sin(x) \)

\(\Large x\) 0 \(\Large \tfrac{\pi}{2}\) \(\Large \pi\) \(\Large \tfrac{3\pi}{2}\) \(\Large 2\pi\)
\(\Large y\) 0 1 0 -1 0

 

Sine function

 

Cosine function

The cosine function describes the ratio between the adjacent side and the hypotenuse in a right-angled triangle.

It is written in this form:

 

$$ \Large y = \cos(x) $$

 

Characteristics:

  • The period is \(2\pi\)
  • The amplitude is 1
  • Starts at \((0,1)\)
  • The curve oscillates between -1 and 1

 

Example: Values for \(\large y = \cos(x) \)

\(\Large x\) 0 \(\Large \tfrac{\pi}{2}\) \(\Large \pi\) \(\Large \tfrac{3\pi}{2}\) \(\Large 2\pi\)
\(\Large y\) 1 0 -1 0 1

 

Cosine function

 

Tangent function

The tangent function describes the ratio between the opposite side and the adjacent side in a right-angled triangle.

It is written in this form:

 

$$ \Large y = \tan(x) $$

 

Characteristics:

  • The period is \(\pi\)
  • No amplitude (the function can become infinitely large or small)
  • Asymptotes at \( x = \tfrac{\pi}{2}, \tfrac{3\pi}{2}, \ldots \)
  • Starts at \((0,0)\)

 

Example: Values for \(\large y = \tan(x) \)

\(\Large x\) -\(\Large \tfrac{\pi}{4}\) 0 \(\Large \tfrac{\pi}{4}\) \(\Large \tfrac{\pi}{2}\)
\(\Large y\) -1 0 1 infinity

 

 

Tangent function

 

Summary

Sine, cosine, and tangent are all trigonometric functions with different properties, but they are interconnected.

 

  • Sine starts at 0 and describes the oscillation up and down
  • Cosine starts at 1 and is shifted compared to sine
  • Tangent has asymptotes and repeats with a shorter period

 

The three functions are often used together to describe angles, motion, and periodic phenomena in mathematics and natural sciences.