Trigonometric functions are mathematical functions that relate the angles of a right triangle to the ratios of its sides. In a right triangle, we have angle theta, the opposite side, the adjacent side, and the hypotenuse. These relationships form the foundation of trigonometry.
The three basic trigonometric functions are sine, cosine, and tangent. Sine is defined as the opposite side divided by the hypotenuse. Cosine is the adjacent side divided by the hypotenuse. Tangent is the opposite side divided by the adjacent side. These ratios help us calculate unknown sides and angles in right triangles.
Trigonometric functions can also be understood through the unit circle. In the unit circle, angle theta corresponds to a point on the circle, where the x-coordinate is the cosine value and the y-coordinate is the sine value. As the point moves around the unit circle, the sine and cosine values change accordingly. The tangent function equals the y-coordinate divided by the x-coordinate.
The graphs of trigonometric functions show their periodic nature. The sine function graph resembles a wave, starting from zero, reaching one at pi over two, returning to zero at pi, dropping to negative one at three pi over two, and back to zero at two pi, repeating this cycle. The cosine function is similar but shifted horizontally by pi over two, starting from one. Both functions are periodic with period two pi.
Trigonometric functions have many important identities and applications. The fundamental Pythagorean identity states that sine squared theta plus cosine squared theta equals one. There are also reciprocal functions: cosecant, secant, and cotangent. These functions are widely used in physics for modeling waves and oscillations, in engineering for signal processing, in navigation and surveying, and in computer graphics for rotations and transformations.