In the vast realm of mathematics, trigonometry stands as a bridge between the celestial and the terrestrial, revealing the harmonies that exist in triangles, waves, and cyclical phenomena. Let us embark on a journey to explore the foundational trigonometric ratios.

### Understanding Sine and Cosine

The primary stepping stones in the river of trigonometry are the sine (sin) and cosine (cos) functions.

Given a right triangle:

• Sine (sin) of an angle θ is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• Cosine (cos) of an angle θ is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

For instance, consider a 3-4-5 right triangle as depicted in the diagram:

In this triangle, for angle B:

$$\sin(B) = \frac{opposite}{hypotenuse} = \frac{3}{5}$$
$$\cos(B) = \frac{adjacent}{hypotenuse} = \frac{4}{5}$$

### Extending to Other Ratios

From sine and cosine, we derive the other four trigonometric ratios:

Tangent (tan) is the ratio of sine to cosine: $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$.
Cosecant (csc) is the reciprocal of sine: $$\csc(\theta) = \frac{1}{\sin(\theta)}$$.
Secant (sec) is the reciprocal of cosine: $$\sec(\theta) = \frac{1}{\cos(\theta)}$$.
Cotangent (cot) is the reciprocal of tangent or the ratio of cosine to sine: $$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$$.

Using our 3-4-5 triangle:

$$\tan(B) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{3/5}{4/5} = \frac{3}{4}$$.
$$\csc(B) = \frac{5}{3}$$.
$$\sec(B) = \frac{5}{4}$$.
$$\cot(B) = \frac{4}{3}$$.