Trigonometric functions are a group of important mathematical functions that describe the relationships between angles and side lengths in right triangles. In a right triangle, we have angle theta, opposite side b, adjacent side a, and hypotenuse c.
The basic trigonometric functions include sine, cosine, and tangent. Sine is defined as opposite over hypotenuse, cosine as adjacent over hypotenuse, and tangent as opposite over adjacent. 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 circle, sine and cosine values change accordingly.
The graphs of trigonometric functions show 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 horizontally.
To summarize what we have learned: Trigonometric functions are fundamental mathematical tools that describe relationships between angles and sides in triangles. The basic functions sine, cosine, and tangent have clear geometric meanings in both right triangles and the unit circle. These periodic functions are essential in many fields including physics, engineering, and signal processing.