Trigonometric functions are important functions in mathematics that describe the relationship between angles and side lengths in right triangles. In a right triangle, we have the angle theta, the opposite side b, the adjacent side a, and the hypotenuse c.
Basic trigonometric functions include sine, cosine, and tangent. The sine function is defined as the opposite side divided by the hypotenuse. The cosine function is defined as the adjacent side divided by the hypotenuse. And the tangent function is defined as the opposite side divided by the adjacent side, which is also equal to sine divided by cosine. These relationships help us calculate other sides of a triangle when we know one angle and one side.
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 along 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 at 0, reaching 1 at pi/2, returning to 0 at pi, dropping to -1 at 3pi/2, and finally returning to 0 at 2pi, then repeating. The cosine function is similar but shifted horizontally by pi/2, starting at 1, going through 0, -1, and back to 1. Both functions are periodic with a period of 2pi.
Trigonometric functions have many important identities. The most fundamental is the Pythagorean identity: sine squared theta plus cosine squared theta equals 1. There are also addition formulas, negative angle formulas, and periodicity formulas. These identities can be understood intuitively through the unit circle and have wide applications in mathematics, physics, engineering, and many other fields.