In terms of sine and cosine, how is tangent (tan) defined?

Tangent is defined as the ratio of the sine of an angle to the cosine of the same angle. This relationship is fundamental in trigonometry and can be expressed mathematically as:

[

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

]

This definition arises from the right triangle definition of trigonometric functions, where tangent represents the ratio of the opposite side to the adjacent side of an angle in a right triangle. Since sine represents the ratio of the length of the opposite side to the hypotenuse, and cosine represents the ratio of the adjacent side to the hypotenuse, their ratio gives us the tangent.

Understanding this definition is crucial as it forms the basis for numerous trigonometric identities and relationships that are frequently used in various applications, including calculus, physics, and engineering.