Trigonometry Concepts with Interactive Diagrams

Welcome to trigonometry! Today we'll learn about the fundamental ratios that form the building blocks of this important mathematical field.Let's start with a right triangle. Every right triangle has one ninety degree angle and another angle we'll call theta.The three sides of a right triangle have special names based on their relationship to theta. The side opposite to theta is called the opposite, the side next to theta is the adjacent, and the longest side is the hypotenuse.The sine of theta is the ratio of the opposite side to the hypotenuse.The cosine of theta is the ratio of the adjacent side to the hypotenuse.And the tangent of theta is the ratio of the opposite side to the adjacent side.To help remember these ratios, we use the mnemonic device SOH-CAH-TOA.These ratios remain constant for similar triangles of different sizes, as long as their angles are the same.No matter how large or small the triangle, if the angles remain the same, the ratios of the sides will be equal.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.As we move through the first quadrant, we'll see how the coordinates relate to sine and cosine.We can measure these angles in both degrees and radians. One hundred eighty degrees equals pi radians.As we complete the circle, notice how the x and y coordinates cycle through their values.These circular motions will help us understand how trigonometric functions create wave patterns.Now we'll see how circular motion creates wave patterns.As a point moves around the unit circle, we can track its vertical and horizontal positions.The sine wave is created by tracking the vertical position of our point.The cosine wave tracks the horizontal position, creating a similar wave shifted by ninety degrees.Let's examine the key features of these waves. The maximum value is positive one, and the minimum is negative one.The waves complete one full cycle every two pi radians. This is called the period.Let's review what we've learned about trigonometric graphs.Thanks for exploring trigonometric functions with Spark.E!