Trigonometric Functions Explained with Animations

Now that we understand the unit circle, let's see how it creates sine and cosine waves.As a point moves around the unit circle, we can track its vertical and horizontal positions.The vertical position creates the sine wave, while the horizontal position creates the cosine wave.The blue curve shows sine of theta, and the green curve shows cosine of theta.Both waves have an amplitude of one, meaning they oscillate between positive one and negative one.The waves repeat every two pi radians, which we call the period.Notice that the cosine wave is shifted one-quarter period, or pi over two radians, ahead of the sine wave.Trigonometric functions appear in many natural phenomena. Let's look at a pendulum's motion.The angle of a pendulum follows a cosine function, creating simple harmonic motion.Sound waves also exhibit trigonometric patterns, often with exponential decay.Now let's explore the tangent function, which is defined as the ratio of sine to cosine.In a right triangle, tangent represents the ratio of the opposite side to the adjacent side.The tangent function has a unique graph with vertical asymptotes at ninety degrees.One practical application of tangent is finding heights of tall objects.By measuring the angle of elevation and distance from the object, we can calculate its height using tangent.