Welcome to trigonometry. Trigonometric functions are important mathematical functions that describe the relationships between angles and sides in right triangles. In a right triangle, we have the angle theta, the opposite side b, the adjacent side a, and the hypotenuse c.
The basic trigonometric functions include sine, cosine, and tangent. The sine function is defined as the ratio of the opposite side to the hypotenuse. The cosine function is defined as the ratio of the adjacent side to the hypotenuse. And the tangent function is defined as the ratio of the opposite side to the adjacent side, which is also equal to sine divided by cosine. These relationships help us calculate the 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, the angle theta corresponds to a point on the circle. The x-coordinate of this point 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 over 2, returning to 0 at pi, dropping to negative 1 at three pi over 2, and finally returning to 0 at two pi, then repeating. The cosine function is similar but shifted horizontally by pi over 2, starting at 1, going through 0 and negative 1, then back to 1. Both functions are periodic with a period of two pi.
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 visually using the unit circle and have wide applications in mathematics, physics, engineering, and many other fields.