Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides. The three primary trigonometric functions are sine, cosine, and tangent, and all of these functions are derived from the right triangle. In this article, we will explore each of these functions and their properties.

The Right Triangle

A right triangle is a triangle with one angle measuring exactly 90 degrees. In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. We can use the lengths of the legs and hypotenuse to define the ratios that form the basis of the trigonometric functions.

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Sine

The sine function is defined as the ratio of the length of the side opposite an angle to the length of the hypotenuse. In other words, for a right triangle with an angle 𝜃, the sine of 𝜃 is given by:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

The sine function is always between -1 and 1. When 𝜃 is 0 degrees, the sine function is 0, and when 𝜃 is 90 degrees, the sine function is 1.

Cosine

The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, for a right triangle with an angle 𝜃, the cosine of 𝜃 is given by:

$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

Like the sine function, the cosine function is always between -1 and 1. When 𝜃 is 0 degrees, the cosine function is 1, and when 𝜃 is 90 degrees, the cosine function is 0.

Tangent

The tangent function is defined as the ratio of the length of the side opposite an angle to the length of the adjacent side. In other words, for a right triangle with an angle 𝜃, the tangent of 𝜃 is given by:

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

The tangent function is not limited to the range between -1 and 1. In fact, it can take on any real value, although it becomes undefined when the adjacent side is 0.

Conclusion

Trigonometric functions are fundamental to a wide range of mathematical and scientific applications, from geometric modeling to the study of waves and oscillations. Understanding the properties and relationships between these functions is critical for anyone studying mathematics, physics, or engineering.

In this article, we have explored the three primary trigonometric functions: sine, cosine, and tangent. By using the ratios of the sides of a right triangle, we can define these functions and use them to solve a wide range of problems.

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