# How is geometry related to trigonometry?

## Trigonometric functions

### Meaning of the trigonometric functions

The three sides of a right triangle are given:

- Edge of the angle \ (\ alpha \): 12 cm
- Opposite cathetus of the angle \ (\ alpha \): 5 cm
- Hypotenuse: 13 cm

The sine, i.e. the ratio of the opposite side to the hypotenuse, can be easily calculated:

\ [\ sin \ alpha = \ frac {\ text {Opposite cathet}} {\ text {Hypotenuse}} = \ frac {5 \ text {cm}} {13 \ text {cm}} \ approx 0.385 \]

Now we know that the sine of the angle \ (\ alpha \) of this triangle is (approximately) 0.385 ... but what does that mean? What have we just calculated?

Let us consider a second example. Then it becomes immediately clear what it is all about.

The three sides of a right triangle are given:

- Edge of the angle \ (\ alpha \): 24 cm
- Opposite cathetus of the angle \ (\ alpha \): 10 cm
- Hypotenuse: 26 cm

In case you don't immediately notice, the sides of this triangle are twice as long as the sides of the first triangle. If you were to draw the two triangles, you might find that although they are different sizes, the three angles match.

We calculate the sine again, i.e. the ratio of the opposite side to the hypotenuse:

\ [\ sin \ alpha = \ frac {\ text {Opposite cathet}} {\ text {Hypotenuse}} = \ frac {10 \ text {cm}} {26 \ text {cm}} \ approx 0.385 \]

Although the two triangles under consideration are of different sizes, the sine of the angle \ (\ alpha \) has the same value!

We know that the following applies: \ (\ sin \ alpha \ approx 0.385 \).

If we solve the equation for \ (\ alpha \), we know how big the angle is:

\ (\ alpha = \ sin ^ {- 1} (0.385) \ approx 22.64 ° \)

The angle functions can be used to calculate the angles of a triangle without having to measure a single angle.

**Notes on calculating with the calculator**

- Your calculator must be set to DEG (Degree).
- The side lengths of the triangle (in our example: opposite cathetus and hypotenuse) must have the same unit - e.g. cm (centimeters) or m (meters).
- To calculate sine (angle \ (\ alpha \) is given), you have to enter the angle in degrees - e.g. 30 ° or 45 °.
- To calculate the angle \ (\ alpha \) (sine is given), you have to use the inverse function of the sine \ (\ sin ^ {- 1} \). There is a corresponding button on your calculator for this.

In the next chapter we will deal with the unit circle. This helps to graphically illustrate the trigonometric functions. We will also see that trigonometric functions are defined for any (positive and negative) angle. So far we have only defined the trigonometric functions using right-angled triangles, which is why our consideration has been limited to angles between 0 ° and 90 °.

### More about trigonometry

You can find more information about trigonometry in the following chapters.

Basics | |

Trigonometric functions | |

Unit circle | |

Trigonometric functions | |

Sine | \ (\ sin \ alpha = \ frac {\ text {Opposite cathet}} {\ text {Hypotenuse}} \) |

Cosine | \ (\ cos \ alpha = \ frac {\ text {adjacent}} {\ text {hypotenuse}} \) |

tangent | \ (\ tan \ alpha = \ frac {\ text {opposite side}} {\ text {adjacent side}} \) |

Reciprocal values | |

Cosekans | \ (\ csc \ alpha = \ frac {\ text {Hypotenuse}} {\ text {Opposite cathet}} = \ frac {1} {\ sin \ alpha} \) |

Secans | \ (\ sec \ alpha = \ frac {\ text {hypotenuse}} {\ text {adjacent}} \ phantom {1 \:} = \ frac {1} {\ cos \ alpha} \) |

Cotangent | \ (\ cot \ alpha = \ frac {\ text {adjacent side}} {\ text {opposite side}} = \ frac {1} {\ tan \ alpha} \) |

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