# Precalculus

## Trigonometry # Trigonometric Functions in a Right Triangle

In this section, we introduce the trigonometric functions using a right triangle. There are $$6$$ main trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. For acute angles, these functions can be defined as ratios between the sides of a right triangle.

Consider a right triangle $$ABC$$ with an acute angle of $$A = \alpha.$$ The side $$b$$ between the angle $$\alpha$$ and the right angle $$C$$ is called the adjacent leg to angle $$\alpha.$$ Respectively, the other leg $$a$$ is called opposite to angle $$\alpha.$$

The sine of $$\alpha,$$ abbreviated $$\sin \alpha,$$ is the ratio of the length of the opposite leg to the length of the hypotenuse:

The cosine of $$\alpha,$$ abbreviated $$\cos \alpha,$$ is the ratio of the length of the adjacent leg to the length of the hypotenuse:

The tangent of $$\alpha,$$ abbreviated $$\tan \alpha,$$ is the ratio of the length of the opposite leg to the length of the adjacent leg:

The tangent function can be expressed in terms of the sine and cosine:

$\tan \alpha = \frac{a}{b} = \frac{a}{c} \cdot \frac{c}{b} = \frac{{\frac{a}{c}}}{{\frac{b}{c}}} = \frac{{\sin \alpha }}{{\cos \alpha }}.$

If we take the reciprocals of the sine, cosine, and tangent functions, we obtain expressions for the other $$3$$ trigonometric functions.

The reciprocal of the sine function is called the cosecant. The cosecant of $$\alpha,$$ denoted $$\csc \alpha,$$ is the ratio of the length of the hypotenuse to the length of the opposite leg:

The reciprocal of the cosine function is called the secant. The secant of $$\alpha,$$ denoted $$\sec \alpha,$$ is the ratio of the length of the hypotenuse to the length of the adjacent leg:

The reciprocal of the tangent is called the cotangent. The cotangent of $$\alpha,$$ denoted $$\cot \alpha,$$ is the ratio of the length of the adjacent leg to the length of the opposite leg:

### Example

The top of the Eiffel Tower is seen from a distance of $$d=500\,m$$ at an angle of $$\alpha = 31^\circ.$$ Find the tower's height.

We know the angle of elevation $$\alpha$$ and the adjacent leg $$d.$$ The tower's height $$H$$ is the opposite leg in the right triangle. Hence,
$H = d\tan \alpha = 500 \times \tan {31^\circ} = 500 \times 0,6 = 300\,m.$