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 an angle theta, the opposite side, the adjacent side, and the hypotenuse. These functions are fundamental tools in mathematics, physics, and engineering.
The cosine function is one of the primary trigonometric functions. In a right triangle, the cosine of an angle theta is defined as the ratio of the adjacent side to the hypotenuse. More generally, in the unit circle, the cosine of an angle is the x-coordinate of the point where the angle intersects the circle.
In the unit circle, which has a radius of one, trigonometric functions have a clear geometric interpretation. The cosine of an angle is the x-coordinate of the point where the angle intersects the circle, while the sine is the y-coordinate. As the angle changes, we can see how the cosine value varies between negative one and positive one.
The cosine function has a distinctive wave-like graph. It has a period of two pi, meaning it repeats every two pi units. The range is from negative one to positive one. Key values include cosine of zero equals one, cosine of pi over two equals zero, cosine of pi equals negative one, and the pattern repeats. This periodic behavior makes trigonometric functions essential for modeling waves and oscillations.