Trigonometry Concepts with Interactive Diagrams

In trigonometry, we start with a right triangle, which always has one 90-degree angle.Let's label the sides of our triangle relative to an angle theta. The side opposite to theta, the side adjacent to theta, and the hypotenuse, which is always the longest side.The sine of an angle is the ratio of the opposite side to the hypotenuse. We remember this as SOH - Sine equals Opposite over Hypotenuse.The cosine of an angle is the ratio of the adjacent side to the hypotenuse. We remember this as CAH - Cosine equals Adjacent over Hypotenuse.The tangent of an angle is the ratio of the opposite side to the adjacent side. We remember this as TOA - Tangent equals Opposite over Adjacent.Let's look at a practical example. How can we find the height of a tree using these ratios?Since we want to find the height, which is the opposite side, and we know the adjacent side, we'll use the tangent ratio. The height equals the adjacent side times the tangent of thirty-five degrees.To understand how sine and cosine waves are formed, we'll start with a point moving around the unit circle.As our point moves around the circle, we'll track its vertical position to create the sine wave.And we'll track its horizontal position to create the cosine wave.These dashed lines will help us track the vertical and horizontal positions of our point.As our point rotates, watch how the sine wave forms from the vertical position, and the cosine wave forms from the horizontal position.The sine wave represents the vertical position of our point as it moves around the circle.While the cosine wave represents the horizontal position of the same point.Notice how the cosine wave is the same shape as the sine wave, just shifted by pi over two radians.Let's watch another complete cycle to reinforce these relationships.Now we'll explore how changing different parameters affects the sine function.First, let's examine amplitude. The amplitude determines how tall or short the waves are.Next, let's look at how changing the period affects the wave's frequency.Finally, let's examine phase shifts, which move the wave left or right.These transformations can be combined to create more complex waveforms.Here's an example with amplitude 2, frequency one-half, and a phase shift of pi over 4.