Trigonometry ratios

Answer 1

Trigonometry ratios are fundamental concepts in trigonometry, which is the study of the relationships between the angles and sides of triangles, particularly right-angled triangles. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan), and they are defined as follows for a given angle \(\theta\) in a right-angled triangle:

1. **Sine (sin)**:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
This ratio compares the length of the side opposite the angle \(\theta\) to the length of the hypotenuse.

2. **Cosine (cos)**:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]
This ratio compares the length of the side adjacent to the angle \(\theta\) to the length of the hypotenuse.

3. **Tangent (tan)**:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
This ratio compares the length of the side opposite the angle \(\theta\) to the length of the side adjacent to the angle.

Besides these primary trigonometric ratios, there are three reciprocal ratios:

4. **Cosecant (csc)**:
\[
\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}
\]

5. **Secant (sec)**:
\[
\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}
\]

6. **Cotangent (cot)**:
\[
\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}
\]

These trigonometric ratios are not only crucial for solving problems involving right-angled triangles but also play significant roles in various applications of mathematics, physics, engineering, and other sciences. Furthermore, the trigonometric functions can be extended to general angles and even complex numbers, providing a broad toolkit for analyzing periodic functions and waveforms, among other uses.