Tools to use Calculus
Trigonometry		"Soh-cah-toa Batman! That's 2000 years."
The vast array of elements and concepts brought together here can be frightening and confusing. Pythagoras (582 BC - 496 BC) known as "the father of numbers" developed the theorem bearing his name. The Pythagorean Theorem states the the sum of the areas squares of the legs of a right triangle equals the area of the square of the hypotenuse. Euclid (365 BC - 278 BC) wrote a set of books called "The Elements" deducing the properties of geometrical objects and integers from a set of integers. Hipparchus (circa 190 BC - circa 120 BC) possibly the greatest astronomical observer of antiquity developed a his numerical trigonometry. Ai-Battani (850 - 929 CE) the greatest Muslim astronomer and mathematician is considered "the father of trigonometry". Niccolò Tartaglia (1499-1557) was the first mathematician to solve the general cubic equation, a3x3 + a2x2 + a1x + a0 = 0 bringing to light i = -1 used in the solution of periodic functions. René Descartes (March 31, 1596 - February 11, 1650) constructed the Cartesian Coordinate Plane.
The values of trigonometric functions repeat at regular intervals. Consider the equation and graph of a circle. This equation does not fit the definition of a function. It defines a relation which can be broken up into several related functions. There are six trigonometric functions, but only two are unique. Sine and Cosine support the elegant simplicity of Trigonometry.
Polar Coordinates
The geometry of the Cartesian Coordinate Plane supports the property of Ordered Pairs. This is not limited to the structure and notation of (x, y). Polar Coordinates possess the same geometry of Cartesian Coordinates. After all this is the geometry of the Cartesian Plane. The (x, y) coordinates are perpendicular distances to a unique point from the Origin. The (r, Θ) coordinate designates each point uniquely as being a distance r from the Origin on a line the makes an angle Θ with a ray running horizontal and to the right of the Origin.
C = 2r		Degrees of Freedom
The circumference of a circle is 2r where r is the radius of the circle. In terms of Polar Coordinates this represents a full rotation of the distance r about the Origin. Start at the point (r, 0), go to the point (r, 2) and stop. The angle Θ, theta, can be measured in degrees. This may momentarily clear things up, but a great amount of simplicity is lost by saying "Start at the point (r, 0°), go to the point (r, 360°) and stop." The native unit of measurement for Θ is the Radian. Degrees, Radians and Arclength An angle measured in degrees give one piece of information. The size of the angle. Multiplying a distance by, say 36° is meaningless. But, take that same distance and multiply it by /5 radians results in the length of that 36° worth of circumference. This is called the arclength. Radian measurements of angles allow for more information to be gathered about a system than does using degrees for about the same amount of calculation effort. Granted, most people can see degrees easier than radians, especially the classics. Forty-five degrees or the 30-60-90 triangle. Its do to how they were raised. The point is that C = 2r and there are 360 degrees in a circle. Additionally, most people would rather have a straight line than a curved piece of circumference.		"What is the degree measurement of 2/3 radians?" "Huh?" "If a full circle has 2 radians; how many degrees is that?" "360°." "And radians is." "180°." "And 1/2 radians?" "90 degrees." "And one fourth radians?" "45 degrees." "So one third radians is. " "One eighty divided by 3 is 60. So, 2/3 radians is 120°." "See, this is why I don't test well. I never memorized these degrees to radians values. I just know that C = 2r and I calculate it out each time I need it."
Anatomy of a Triangle
The hypotenuse is labeled h. The side opposite angle Θ is labeled o and the other side, adjacent to Θ, is labeled a. The two functions sin(Θ) and cos(Θ) return the length of the side opposite Θ and side adjacent Θ divided by the length of the hypotenuse for any given measure of the angle.
Sine and Cosine		Θ can overlap on itself
The input for both sine and cosine is Θ which is an angle measured in either degrees or radians. A quarter turn counterclockwise is 90° or /2 radians. Three quarters the way around is 3/2 radians and a full turn is 360°. Keep turning and the angle exceeds 2 radians. When turning clockwise Θ is negative. Thus, D = - < Θ < defines the domain for both functions. The outputs of sine and cosine are the ratios of the length of one leg of the triangle to its hypotenuse. Sine is the ratio of the opposite side over the hypotenuse and cosine is the ratio of the adjacent side over the hypotenuse. The range for both sine and cosine is -1 ≤ T ≤ 1. Sine is Opposite over Hypotenuse. Cosine is Adjacent over Hypotenuse.		sin(Θ)
		cos(Θ)
Sine and Cosine are Periodic Functions
The definitions of sine and cosine are repeated on this page to reinforce the reality that their values repeat at the regular interval of 2 radians. These functions connect a triangle, polar coordinates and cartesian coordinates. sin(Θ) =  o  and  cos(Θ) =  a h h Substitute radius for hypotenuse sin(Θ) =  o  and  cos(Θ) =  a r r The opposite side of the triangle is the vertical distance from the Origin, so substitute y for o sin(Θ) =  y  and  cos(Θ) =  a r r The adjacent side of the triangle is the horizontal distance from the Origin, so substitute x for a sin(Θ) =  y  and  cos(Θ) =  x r r Multiply both side of each equation by the r rsin(Θ) = y and rcos(Θ) = x which gives a nice connection between polar coordinates and cartesian coordinates through a triangle.
Something of Value - section six		Tools to use Calculus - section topics