The Special Circle

Unit Circle : circle centered at the origin, and a radius of 1 

x² + y² = 1

Since the radius in the unit circle is one and s = r (remember s is arc length and r is radius) r = 1, meaning that in a unit circle = s.

To complete the unit circle it is important to remember our special right triangles: 30-60-90, 45-45-90

Their sides are related like this:

 

 

 

Once you remember that, it is easy to go on, simply drop an altitude from an point in the first quadrant, I will begin with the point -

 

 

Dropping an altitude from creates a 30 - 60 - 90 triangle ( = 30). Since the hypotenuse equals 1 (because it's the radius), we saw above that the base(x) will equal half of that, so (1/2) and the height(y) will equal  or .

 

It would look like this:

 

 



 

Next, is the point , dropping an altitude form here would create a 45 - 45- 90 triangle  (  = 45).

Again the hypotenuse it 1 because the radius is one meaning the both of the sides would equal but in mathematics we rationalize that into , both x and y would equal this.

 

This next point would look like this:

 

 

The last point is easiest because technically we've already done it. When you drop and altitude from, it also creates a 30 - 60 - 90 triangle with a hypotenuse of one; however, this time the legs (x and y) have been switched.

 

The base (x) is now half the hypotenuse  times the square root of three:. And the height is half the radius so .

 

It looks like this:





This is the Unit Circle, the rest is completed by manipulating this same number depending on which axis the first quadrant is reflected. The end product look like this:

 

 

Trig Functions:

 

Dividing two of the three sides of a right triangle will warrant a special result, no matter which sides are divided.

 

 

The functions Cosine (adjacent over hypotenuse) and Secant (hypotenuse over adjacent), are both even functions. This means that the function of -, will yield a result  (the function being cosine or secant).

When graphed, these function are also reflections over the y - axis.

 

 

The functions Sine(opposite over hypotenuse), Cosecant (hypotenuse over opposite), Tangent (opposite over adjacent), Cotangent (djacent over opposite) are all odd functions. This means that the function of - will yield a result of the -f().

When graphed, functions are also reflected over the x - axis.





Radian and Degree Measure

Trigonometry: The measurement of triangles with the relationship among sides and angles.

Angles: 2 rays with a common end point (vertex)

The angle is measured by the amount of rotation from the initial side to the terminal side. If measured in a counterclockwise direction the measurement is positive and if it is measured in a clockwise direction the measurement is negative.

If two angles in standard position have the same terminal side, then they are called coterminal angles.

Two positive angles  A and B are complementary if their sum is:

Two positive angles A and B are supplementary if their sum is:

Converting Degrees to Radians:

Example:

The radian measure formula can be used to measure arc length along a circle

Example:

 

First convert 115 degrees to radiant

Next plug in information in to the radian measure formula

Estimate your answer

Solving Exponential and Logarithmic Equations

There are a few strategies for solving exponential and logarithmic equations:

1. Rewrite the given equation in a form to use the One-to-One Properties of exponential or logarithmic functions. 
2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. 
3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of logarithmic functions. 

One-to-One Properties: 

 if and only if 

and 

if and only if 

Inverse Properties:

Logarithmic equations can also be manipulated by exponentiation.

Notice that: 

•  is not a solution as it is extraneous 
• Be sure to check for extraneous solutions
• You can also check for extraneous solutions by graphing 

• Exponentiation is executed after the logarithmic expression has been isolated 

Investment: 

It's important to remember the formula for continually compounded interest: 

where A is the final balance, P is the principal, r is the rate and t is the time

Tips: 

• To solve exponential equations, it is useful to first isolate the exponential expression, then take the logarithm of both sides and solve for the variable. 
• To solve logarithmic equations, condense the logarithmic part into a single logarithm, then rewrite in exponential form and solve for the variable.