4.3 Right Triangle Trigonometry

Trigonometric Functions

In any right triangle, all three sides have a relation with any of the 3 angles. Relative to angle x, these three angles are either adjacent to angle x, opposite of angle x, or the hypotenuse of the triangle.

Using this principle, 6 different ratios can be created. It is also important to note that angle x is greater than 0 degrees and less than 90 degrees.

SOH-CAH-TOA

                      

                      

The functions cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent respectively. These are known as Reciprocal Identities.

                             

The cotangent function and tangent are known as Quotient Identities

       

This information, along with the Pythagorean Theorem can be used to to find the Pythagorean Identities:

          

Lets derive the first identity:  by using the Pythagorean theorem and our knowledge of the trigonometric functions. 

           

           

Trigonometry and the Unit Circle

Looking at the first quadrant of the unit circle, imagine that a right triangle is created with an angle measure of  at the origin. Using the property triangles, it can be inferred that the remaining angle is . By using the trigonometric functions, either leg of the triangle can be determined. To find the leg opposite of angle x, you can use the sine function to determine the side length. 

     Because we know that angle x is , we replace x with the degree of the angle. Since the the radius of the unit is 1 unit, we can say that the hypotenuse of the triangle is 1. 

      Here we can see that the sine of  is equal to the side opposite of the angle. This  value also corresponds to the y value of the point that lies on the unit circle. Therefore,  is equal to the y value of the point created where the hypotenuse, and the edge of the circle intersect. 

Using the same logic, it can be seen the the  is equal to the side adjacent to angle x. Therefore,  is equal to the x value of the point created where the hypotenuse and the edge of the circle intersect.

This information can be used to find any point on the unit circle if the corresponding angle is known.

                                         

4.1 Radian and Degree Measure

What is an angle?

An angle is simply two rays that share a common endpoint. An angle is a vector meaning it has magnitude and direction. The starting position of a ray is refereed to as the initial side of the angle and the position remaining after rotation is refereed to as the terminal side. The common end point where the rays meet is the vertex.

Envisioning a coordinate system, an angle is in standard position if the initial side of the angle lies on the positive x-axis.

Positive angles result from counterclockwise rotation and negative angles result from clockwise rotation.








Angles are often labels with upper case or Greek letters. Angles are coterminal if they share a common initial and terminal side.

In this case, for all three angles created by the blue, red, and green lines, the resulting angles are coterminal.

What are we measuring when we measure an angle?
To measure an angle means to measure the amount of revolution. 

Since the circumference of a circle is 2πr, a central angle containing one full revolution counterclockwise creates an arc length s, equal to 2πr. This means that 2π radians correspond to 360° and π radians correspond to 180° and so on. 

If one circle (360°) is also 2π radians, there must be about 6.26 radians in one full circle. 

A circle can be divided and measured the following ways with the Unit Circle:

What are radians?
Radians are another way to measure angles besides degrees. Envision a central angle θ on a circle centered around the origin showing a vertex at the origin. A single radian is the amount of rotation needed to make the length of the intercepted arc s equal to the radius r.
If s = r then, θ =1 radian.









To determine arc length a proportion can be used.

Where...
Angle Measure: θ
Total Measure of One Circle: 360 (in degrees)  or 2π (in radians)
Arc Length: s (if degrees or radians are used for the total measure of one circle that same unit will remain for arc length)
Circumference: 2πr

Similarly, to convert from radians to degrees and vise versa a proportion is used.

Multiply the magnitude by... 

to go from degrees to radians and... 

to go from radians to degrees. The original unit will cancel out and the desired unit will remain.

3.3 Properties of Logarithms

Logarithmic Properties

If is a positive number not equal to 1, is a real number, andandare positive real numbers then the following properties are true:

Property 1:

Example:

Property 2:

Example:

Property 3:

Example:

These properties mentioned above are also true for natural logarithms,, under the same conditions, so can be substituted with if needed. These properties can help us to expand and condensing logarithmic expressions.

Expanding Logarithmic Expressions

Expand this expression:

Property 3

Property 2

Property 1

Condensing Expressions

Condense this expression:

   Property 3

Property 1

Property 2

Change-of-Base Formula 

                                        

If you are trying to solve for logarithms that have bases other than those on your calculator (andare the most common on calculators) ,you can use the change -of –base formula to solve for them. 

If  and are positive real numbers and andthen you can change the base ofwith any of these given formulas:

                                                             

    Base changed to b

  Base changed to 10

        Base changed to e

Example:

Base change to 10

               

               

- This process would be the same if you wished to change the base to e or any other positive real number, just replace 10 with the value you want.