# 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 A = α. The side b between the angle α and the right angle C is called the adjacent leg to angle α. Respectively, the other leg a is called opposite to angle α.

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

$\sin \alpha = \frac{a}{c} = \frac{\text{opposite leg}}{\text{hypotenuse}}$

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

$\cos \alpha = \frac{b}{c} = \frac{\text{adjacent leg}}{\text{hypotenuse}}$

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

$\tan \alpha = \frac{a}{b} = \frac{\text{opposite leg}}{\text{adjacent leg}}$

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:

$\csc \alpha = \frac{1}{\sin\alpha} = \frac{c}{a} = \frac{\text{hypotenuse}}{\text{opposite leg}}$

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:

$\sec \alpha = \frac{1}{\cos\alpha} = \frac{c}{b} = \frac{\text{hypotenuse}}{\text{adjacent leg}}$

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:

$\cot \alpha = \frac{1}{\tan\alpha} = \frac{b}{a} = \frac{\text{adjacent leg}}{\text{opposite leg}}$

### 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.

Solution.

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,

$H = d\tan \alpha = 500 \times \tan {31^\circ} = 500 \times 0,6 = 300\,m.$

See solved problems on Page 2.