Trigonometry Concepts with Interactive Diagrams

Welcome to trigonometry! Today we'll learn about the basic trigonometric ratios with Spark.E!Let's start with a right triangle. Every right triangle has one 90-degree angle and another angle we'll call theta.The three sides of a right triangle have special names in relation to an angle. The side opposite to our angle theta is called the opposite side.The side next to our angle is called the adjacent side.And the longest side, opposite to the right angle, is called the hypotenuse.These sides form the basis for our three main trigonometric ratios. Let's start with sine.Sine is the ratio of the opposite side to the hypotenuse.Cosine is the ratio of the adjacent side to the hypotenuse.And tangent is the ratio of the opposite side to the adjacent side.These ratios remain constant for similar triangles of different sizes.No matter how big or small the triangle is, as long as the angles are the same, the ratios of the sides will be equal.To help remember these ratios, we use the mnemonic device SOH-CAH-TOA.SOH stands for Sine equals Opposite over Hypotenuse.CAH stands for Cosine equals Adjacent over Hypotenuse.And TOA stands for Tangent equals Opposite over Adjacent.Now that we understand the basic ratios, we're ready to explore how they relate to the unit circle.The unit circle is a circle with radius 1 centered at the origin.A point moving around this circle traces out different angles from the positive x-axis.The coordinates of this point give us the cosine and sine values for each angle.We can also measure angles in radians, which is often more useful in mathematics.As we complete the circle, we return to our starting point, having traced out two pi radians or three hundred and sixty degrees.Now let's see how circular motion creates wave patterns.As a point moves around the unit circle, we can track its vertical and horizontal positions.The height of the point creates the sine function, while its horizontal position creates the cosine function.Let's examine the key features of these waves.The period of both sine and cosine is two pi, meaning they complete one full cycle every two pi units.These wave patterns appear throughout nature and technology.To summarize what we've learned about trigonometric functions.Thanks for exploring trigonometric functions with Spark.E!