Law of sines and cosines

The law of sines is sin(A)/a=sin(B)/b. It is used when you have two angles and one side. The law of cosines is c^2=a^2+b^2-2abcos(C). This is used when you have three sides or two sides and an angle. Both of these are used to find the angles and sides of non-right triangles. Each law can be used in terms of a, b, or c.

Verifying trig identities

Essentially to verify a trig identity you must use different forms of substituting trig functions. One major one is tan=sin/cos. Another common one is sin^2+cos^2=1. 1+tan^2=sec^2 and 1+cot^2=csc^2 are also very commonly used. In many ways, verifying trig identities is like solving a puzzle. You must chose the most complicated side to work one, and use only that side. Most of the times you start by putting everything in terms of sin and cos.

Tangent

Tangent is a trigonometric function. Tangent can also be graphed. When trying to find an angle of a right right triangle, you would use tangent by saying tan(angle)=opposite/adjacent. The graph of tangent is not continuous. No matter what transformation the tangent graph goes through, there will always be infinite zeros. There are various trig identifies concerning tangent, including sum/difference, half angle, and double angle identities. Tangent is equal to sine over cosine. 

Sine and cosine

Sine and cosine are trigonometric functions. They both can be graphed. When trying to find an angle of a right right triangle, you would use sine by saying sin(angle)=opposite/hypotenuse. If you were to use cosine, you would say cos(angle)=adjacent/hypotenuse. Both sine and cosine's graphs are very similar, except that sine starts at x=0 while cosine starts at the high test point of the graph's period. There are various trig identifies concerning sine and cosine, including sum/difference, half angle, and double angle identities.

Rational functions

A rational function is pretty much just a function that contains a ratio of two polynomials. In these functions, there is one number divided by another, thus creating a ratio. An example of a rational function is (x^2 + 3)/(x + 1). Ratios are commonly expressed in fractions, hence the name rational. When graphed, rational functions are discontinuous.

Zeros of a Function

The zeros of a function are where the graph touches the x-axis. Zeros are also called roots or x's. To find a function's zero you have to solve for x by setting the equation to 0.  Basically, the zero of a function is the point at which no matter what number you input, the output will always be 0. Many graphs, like the sin, cos and tan graphs, have infinite zeros, as they continually go above and below the x-axis. Page functions zero is essentially f(x)=0. However, not every function has a zero.  If the vertical shift is higher than the amplitude, then the graph will never touch the x-axis.

Piecewise Functions

A piecewise function is one that when graphed, shows up in pieces. These functions usually have more than one function within them, consisting of two or more equations with restrictions on the size of x. For example, a piecewise function might say f(x)={x^2 if x<4; 5 if x=4; 8-x if x>4. In this example, there are essentially three mini functions that make up the function as a whole. You would use the function that covers the number you are plugging in. If one were to use f(6) in the above function, they would have to use 8-x because >4. In a piecewise function, there are always holes in the graph due to the multiple functions within it.

What is a Function?

A function is essentially an equation that you put one number into, and another comes out. Functions are commonly labeled as f(x). They also usually include an x that is substituted with the number you are putting through the function. Functions can be graphed as well. In every function, each output has a single input. An example of a function would be f(x)=2x+3.