Welcome to trigonometry! Today we'll explore the three fundamental trigonometric functions: sine, cosine, and tangent. These functions relate angles in a right triangle to the ratios of its sides. Sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. These simple ratios are the foundation for understanding periodic phenomena in mathematics, physics, and engineering.
The unit circle is a powerful tool for understanding trigonometric functions. It's a circle with radius one, centered at the origin. For any angle theta measured counterclockwise from the positive x-axis, the point where the angle intersects the circle has coordinates cosine theta and sine theta. The x-coordinate gives us the cosine value, and the y-coordinate gives us the sine value. Watch as the point moves around the circle and see how the sine and cosine values change.
Now let's see how trigonometric functions look as graphs. The sine function starts at zero, rises to one at pi over two, returns to zero at pi, drops to negative one at three pi over two, and completes its cycle at two pi. The cosine function starts at one, drops to zero at pi over two, reaches negative one at pi, returns to zero at three pi over two, and back to one at two pi. Both functions repeat this pattern every two pi units, making them periodic functions.
The tangent function is quite different from sine and cosine. Since tangent equals sine divided by cosine, it becomes undefined whenever cosine equals zero. This happens at pi over two and three pi over two, creating vertical asymptotes where the function approaches positive or negative infinity. Unlike sine and cosine, tangent has a period of pi, not two pi, and its range includes all real numbers. The function passes through the origin and repeats its pattern every pi units.
To summarize what we've learned about trigonometric functions: Sine, cosine, and tangent are fundamental functions that relate angles to triangle side ratios and unit circle coordinates. Sine and cosine have a period of two pi, while tangent has a period of pi. These functions are essential for modeling waves, oscillations, and periodic phenomena in mathematics, physics, and engineering applications.