The foundation of all science and technology is mathematics. One of the most important branches of mathematics is trigonometry. Without trigonometry, our knowledge of physics and engineering would crumble. Suppose we have a circle. Suppose we place this circle around the center of a graph. Suppose that the radius of this circle is 1. Let’s give the name “θ” to the angle between this line and the X axis. Let’s always measure θ from the X axis in the counterclockwise direction. We can measure θ in degrees, or we can measure it based on the distance travelled around the circle. If we travel all the way around the circle, the distance traveled is equal to the circle’s circumference, which is π multiplied by the diameter of 2. When we measure θ based on the distance travelled around the circle, we say that we have measured θ in radians. For each location on the graph, the green ball travelling around the circle has an X coordinate. Let’s give the X coordinate a name and call it the “cosine.” For every value of θ, we will refer to the X coordinate as the “cosine of θ.” Now let’s consider the Y coordinate of the green ball. Let’s also give the Y coordinate a name, and call it the “sine of θ.” The sine and the cosine of an angle can range anywhere between -1 and +1. Consider the triangle that is made with the X axis. The lengths of the sides of this triangle are the sine and the cosine of θ. As the triangle grows, all the lengths of the triangle grow by the same proportion. Most calculators have buttons labeled sine and cosine. You can enter the values of an angle into a calculator, either in degrees or in radians. The sine and cosine buttons will then tell you the values of the sine and the cosine of that angle. Most calculators also have buttons for the “inverse sine” and the “inverse cosine.” In these cases, you enter the value for the sine or the value for the cosine, and the calculator will tell you what angle would have produced that value. Sine and cosine are two of the most important functions in mathematics. They allows us to solve the most critical problems in all branches of physics. Consider flying an aircraft. Suppose we know the X, Y, and Z coordinates of where we want to fly to. The “inverse sine” or the “inverse cosine” button on a calculator will allow us to calculate the angle we should fly at. This will allow us to navigate in all three dimensions. So far, all the triangles we have encountered have had a 90 degree angle. Now, suppose that this angle is a value other than 90 degrees. Correction: A^2 + B^2 = C^2 + 2ABcos(Ψ) This relationship is always true for all triangles. Correction: A^2 + B^2 = C^2 + 2ABcos(Ψ) If this angle is 90 degrees, then the cosine of 90 degrees is zero. A^2 + B^2 = C^2 + 2ABcos(Ψ) The Pythagorean Theorem is actually just a part of this more general equation, which we call the “Law of cosines.” Correction: A^2 + B^2 = C^2 + 2ABcos(Ψ) Just as there is a “Law of Cosines”, there is also a “Law of Sines.” The Laws of Sines is also always true for all triangles. We have the following four relationships. Let us give each of these relationships a name, and we will call them the “Tangent”, the “Cotangent”, the “Secant”, and the “Cosecant.” We started out by saying that θ is the angle from the X axis in the counterclockwise direction. If we go around the circle more than once, the angle can be greater than 360 degrees. If we go around the circle in the opposite direction, the angle will be negative. We also said that instead of measuring θ in degrees, we can measure θ based on the distance travelled around the circle. The circumference of our circle is π multiplied by its diameter of 2. If we travel only half way around the circle, θ is half this value. If go around the circle in the clockwise direction, θ will be negative. If we go around the circle in the clockwise direction twice, the angle will be negative 2 π multiplied by two. We have an X axis and a Y axis. Let us also create an axis that represents the angle θ. The angle θ can range anywhere between negative infinity and positive infinity. For each value of θ, the X coordinate represents the cosine of θ, and the Y coordinate represents the sine of θ. The green line rotates by the angle θ. The red line represents the X coordinate. The blue line represents the Y coordinate. Any function of any shape can be created by adding together sine and cosine waves of different frequencies and amplitudes. By understanding how physical objects respond to sine and cosine waves, we can determine how they will respond to every possible function in the Universe.
More Sonic music @ 12:17 (Green hill + Marble Zone remix).