Trigonometric functions describe the relationship between angles and the sides of a right triangle. The Pythagorean Theorem describes the relationship between the lenghts of the sides of a right triangle.
The unit circle has a radius (c) = 1 and is centered on the origin (0,0).
In a unit circle the x and y coordinates of a point on the circle are the cosine and sine of the angle since c = 1.
Drawing a right triangle inside of the unit circle allows the Pythagorean Theorem to be expressed in this form:
Once we have this pythagorean identity we can derive the following:
Odd functions exhibit this property: f(-x) = -f(x)or f(x) = -f(-x). Sine is an odd function as seen in this example: sin(π/6)= -sin(-π/6).f(-x) = f(x), By using the definition f(-x)=f(x),cosine is an even function. For example cos(-π/6)=cos(π/6).
From this definition, the following relations hold.
Sometimes it is useful to calculate trigonometric functions where the argument is the sum of two addents with the following rules: sin(θ+φ) = sinθcosφ + cosθsinφ. This relationship is demonstrated in this figure:
The double angle identities are just a specific case of the sum of two angles (the case where θ and φ equal one another). They are written compactly as:
The second half angle identity can be obtained by manipulating one of the double angle identities:
Using the previously derived identity and recalling the Pythagorean Theorem inside the unit circle, we can arrive at another half angle identity:
We derive the next half angle identity starting with the relationship between tangent, sine, and cosine, manipulating this relationship, then applying both double angle identities.
The derivation for all the identities follows a similar path to the one below.
Derivatives are functional operators used to determine the rate of change of a function compared to its input as defined by this formula:
Because derivatives are fundamental to calculus, several theorems have been developed as shortcuts to decrease the amount of basic mathematics. Three of the common theorems have been listed below with examples for reference.
The product rule allows derivatives to be taken easily when two functions are multiplied together.
Below is a proof of the rule and step by step explanations.
The quotient rule allows a derivative to be easily calculated when two functions are contained within a fraction.
The chain rule provides a simple way to evaluate derivatives when one function is contained within another funtion.
Integration by parts is a convienient method to evaluate certain integrals that would otherwise be too difficult to evaluate analytically. In many cases, several iterations of the process may have to be performed before the final integral is simple enough to complete without the aid of a computer.
The logic behind this method is simple and can be seen in the derivation below. It stems from the product rule and only applies to functions that are themselves made up of a product of two smaller functions containg the same independent variable. Definite and indefinite integrals are performed the same way.
The example problem demonstrates an integration that requires multiple iterations before reaching the final conclusion.
U substitution is a technique used to integrate functions that, at first glance, seem very difficult to integrate. Essentially, it is the chain rule working backwards. It is used to calculate integrals which contain the derivative of a function, g'(x), and that function, g(x). To use u substitution you must first decide which factor to set equal to u: . The next step is to calculate the derivative of that function as demonstrated in this equation After that, substitute u and du back into the integral and solve.
Once the integral has been evaluated, replace u with the original function to finalize the answer.
Ex. What is the integral of the following function?
There are many instances in which, for one reason or another, a radical will appear within an integral. While in some cases these may not cause much of an issue and can be easily worked around using a u substitution or another method, a majority of the time the solution is more complicated. Another common trick used to solve an otherwise unsolvable integral is a trigonometric substitution. In these problems, manipulation of the radical may be required to convert the integrand into a usable form. Once in the correct layout, a substitution may be made to replace the current variable with a trigonometric function such as:
An example has been worked out using the sine substitution to demonstrate the method. The other two substitutions follow the same method, but are merely used in their respective situations given above.
A series is a sequence of numbers in which every term is added together to produce a final sum. There are many series in mathematics that occur regularly and have been reduced to a simple formula to calculate the entire sum. This makes the math much less tedius, quicker, and easier to evaluate.
A geometeric series is a series in which some number, a, is multiplied by the same ratio in every term. Such a series can be evaluated using the formula:
Another common use of this series is when |c|<1, since . When this occurs, the formula slightly changes to:
It may be easier to understand this if you understand where these formulas comes from.
Ex. What is the geometric sum if c =-0.25?
Notice that it would takes a long time and a lot more work to get the left side of the equation to converge on the answer, whereas the right side, or closed form, of the equation gives you the exact answer immediately.
Another useful series that occurs regularly is the sum of the integers. In this series every integer is added in consecutive order until the nth integer is reached. (ie: 1+2+3...) Its formula is below along with the shortcut to the final summation.
To fully understand where the final summation formula comes from it can be helpful to walk through a step by step derivation of the process.
As an example let's say you wanted to add up the numbers from 1 to 100. Instead of adding each on individually, 1+2+3..., you can use the formula to reach the final answer much quicker.
As you can see using the closed form is a faster and easier method of finding the final answer.
A Taylor series is used to describe a function in a series of the function's derivatives about a point a. It is particularly useful for approximating functions that would otherwise be difficult to solve analytically such as a multipole expansion or difficult integrals. The series can be written as:
To derive the Taylor series first assume that f(x) has a power series expansion about a. Next assume that f(x) is smooth around the neighborhood of a
An example of using the Taylor series is below:
Permutations describe the different possibilities for the order of some set of things. One example of the use of permutations is the question: "In how many different orders can I arrange my three blocks? The answer is six because you can choose any block for the first spot, then you are left with two to choose from and there is only one block remaining for the final spot. Following this pattern for any amount of blocks, the number of possible arrangements for a set of blocks is determined by calculating the factorial for number of blocks:
Suppose we now have ten blocks, but we only wish to arrange three of the blocks. Now there are ten blocks to choose from for the first spot, nine for the second spot, and eight for the third spot. Therefore the number of possibilities is:
In order to create a formula for determining the number of permutations we recognize that:
Example Problem: You and a friend want to play a game of nine ball, so you approach a pool table and begin randomly picking out balls from the pockets. What is the probability that you and your friend pick out the balls numbered one through nine in order? (Consider 16 balls total to include the cue ball).
Combinations are very similar to permutations, but combinations are order independent. Suppose we have a bag of blocks. Combinations tell us the number of possibilities that can occur if we pick a specific number of blocks. In this example we have four blocks to choose from and we will be picking two. As you can see there are six possibilities:
The number of combinations is obtained with a formula similar to the permutation formula but is divided by k! to account for the fact that we are not concerned with the order of our selections when we use combinations:
Example Problem: In a game of poker, what is the probability of being delt a royal flush in spades? What is the probability of being delt any royal flush?
The normal distribution is useful in probability and statistics because it describes a multitude of phenomena that we can study, from people's heights to errors in machine parts manufacturing. The normal curve has a characteristic bell shape and can be modified with two parameters: mean (μ) and standard deviation (σ). The standard normal distribution has a mean of 0 and a standard deviation of 1:
Below are examples of modifying the mean and standard deviation of the normal curve. In order to shift the distribution from side to side, we change the mean. In order to make the distribution "bulge out" we can increase the standard deviation.
1 Smail, Lloyd L. Trigonometry: Plane and Spherical. New York: McGraw-Hill, 1952. 24-25.
2 Ibid. 94-95.
3 Ibid. 126-44.
4 Ayres, Frank, Jr. Schaum's Easy Outlines: Calculus. Fifth Edition. New York: McGraw-Hill, 2009. 73.
5 Ibid. 83.
6 Ibid. 80.
7 Ibid. 259-60.
8 Ibid. 266.
9 Ibid. 268-70.
10 Ibid. 360-61.
11 Ibid. 396-397.
12 Devore, Jay L. Probability and Statistics for Engineering and the Sciences. Belmont: Thomson Higher Education, 2008. 62-65.
13 Ibid. 145-51.