Trigonometric functions are fundamental mathematical tools that describe the relationships between angles and side lengths in right triangles. In a right triangle, we have angle theta, the opposite side b, the adjacent side a, and the hypotenuse c.
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 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 reveal 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 have 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 one. There are also addition formulas, negative angle formulas, and periodic formulas. These identities can be understood intuitively through the unit circle and have wide applications in mathematics, physics, and engineering.