Trigonometric Functions: Graphs Made Easy

Welcome to understanding the basic sine wave, the foundation of trigonometry!Let's start by looking at how the sine wave is created from circular motion.Imagine a point moving around a circle. As it rotates, we can track its height above or below the center.As our point rotates around the circle, it traces out a sine wave when we plot its height against the angle of rotation.Let's look at some key points on our sine wave. At zero, the sine is zero.At pi over two, or ninety degrees, sine reaches its maximum value of positive one.At pi, or one hundred eighty degrees, sine returns to zero.At three pi over two, or two hundred seventy degrees, sine reaches its minimum value of negative one.Finally, at two pi, or three hundred sixty degrees, we complete one full cycle and return to zero.The sine wave has two important properties: its period and amplitude.The period is two pi, meaning the wave repeats every two pi units. The amplitude is one, showing how far the wave extends above and below the x-axis.Now that we understand the basic sine wave, we're ready to explore its relationship with the cosine function.Now that we understand the sine wave, let's explore its close relative: the cosine wave.First, let's recall our sine wave in blue.Now, let's add the cosine wave in red. Notice how it has the same shape as sine, but starts at a different point.Let's examine some key points of the cosine function. At x equals zero, cosine starts at its maximum value of one.At x equals pi over two, cosine crosses zero, exactly where sine reaches its maximum.At x equals pi, cosine reaches its minimum of negative one.And at three pi over two, it returns to zero again.The cosine wave is actually the same as the sine wave, just shifted left by pi over two units.Notice how sine and cosine complement each other. When one function is at zero, the other is at either positive or negative one.We'll now explore how to transform trigonometric functions by modifying their equations.First, let's look at amplitude. When we multiply sine by a number A, it changes the height of the wave. Here, A equals 2 doubles the height.Next is frequency. When we multiply x by B inside sine, it changes how quickly the wave repeats. B equals 2 makes it complete two cycles in the space of one.Phase shift moves the wave horizontally. Subtracting C inside sine shifts the wave right by C divided by B units. Here we shift by π over 4 units right.Adding D at the end shifts the entire wave up or down. Here, adding 1 moves everything up one unit.Let's review how each parameter affects the wave:We can combine all these transformations. Here's a wave with amplitude one point five, frequency three, phase shift pi over three, and shifted down by zero point five units.