Trigonometric functions are mathematical functions that describe the relationships between angles and side lengths in triangles. In a right triangle, we have angle theta, opposite side b, adjacent side a, and hypotenuse c. These functions are essential tools in mathematics, physics, and engineering.
The three basic trigonometric functions are sine, cosine, and tangent. Sine is the ratio of the opposite side to the hypotenuse. Cosine is the ratio of the adjacent side to the hypotenuse. Tangent is the ratio of the opposite side to the adjacent side. These ratios help us calculate unknown sides when we know one angle and one side.
The cosine function is one of the fundamental trigonometric functions. In a right triangle, cosine of an angle is the ratio of the adjacent side to the hypotenuse. In the unit circle, cosine represents the x-coordinate of the point where the angle's terminal side intersects the circle. The cosine function has a range from negative one to positive one, a period of two pi, and is an even function.
The graphs of trigonometric functions reveal their periodic nature. The sine function starts at zero, reaches one at pi over two, returns to zero at pi, goes to negative one at three pi over two, and completes the cycle at two pi. The cosine function follows a similar pattern but is shifted by pi over two. Both functions repeat this pattern every two pi units, making them essential for modeling periodic phenomena in physics and engineering.
Trigonometric functions have countless applications across science and technology. The fundamental Pythagorean identity shows that sine squared plus cosine squared equals one. These functions are essential in physics for describing wave motion and oscillations, in engineering for signal processing, in navigation systems like GPS, and in computer graphics for rotations. Understanding trigonometric functions opens doors to advanced mathematics and practical problem-solving in many fields.