Trigonometric functions

 Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of its sides. These functions have widespread applications in various fields, including mathematics, physics, engineering, computer science, and more. The main trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. They can be defined using a right triangle or a unit circle.


Sine (sinθ): The sine of an angle θ is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. Alternatively, in the unit circle, it is the y-coordinate of the point on the unit circle corresponding to the angle θ.

sinθ = (opposite side) / (hypotenuse)


Cosine (cosθ): The cosine of an angle θ is defined as the ratio of the length of the adjacent side to the angle to the length of the hypotenuse in a right triangle. In the unit circle, it is the x-coordinate of the point on the unit circle corresponding to the angle θ.

cosθ = (adjacent side) / (hypotenuse)


Tangent (tanθ): The tangent of an angle θ is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side in a right triangle. It can also be represented as the sine divided by the cosine of the same angle.

tanθ = (opposite side) / (adjacent side) = sinθ / cosθ


Cosecant (cscθ): The cosecant of an angle θ is the reciprocal of the sine of that angle.

cscθ = 1 / sinθ


Secant (secθ): The secant of an angle θ is the reciprocal of the cosine of that angle.

secθ = 1 / cosθ


Cotangent (cotθ): The cotangent of an angle θ is the reciprocal of the tangent of that angle.

cotθ = 1 / tanθ = (adjacent side) / (opposite side) = cosθ / sinθ


Trigonometric functions are periodic, meaning their values repeat in regular intervals as the angle increases or decreases. The sine and cosine functions have a period of 360 degrees (2π radians) or any multiple thereof, while the tangent, cotangent, secant, and cosecant functions have a period of 180 degrees (π radians) or any odd multiple thereof.


These trigonometric functions are essential for solving various problems involving angles, distances, and periodic phenomena. They form the foundation of trigonometry and are widely used in calculus, differential equations, and other advanced mathematical subjects

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