10)+Given+a+special+angle+(30,45,60+multiples),+calculate+the+six+trigonometric+ratios+as+simplified+radicals

By Melissa Soule

Background:
The reason that I chose this topic is because in order to understand most of trigonometry it is necessary to understand how to form a unit circle. This topic I found was a struggle in the beginning of its introduction in class however, after time it has become second nature. I also chose this topic because it tests my ability to explain the unit circle and expand my comprehension of the unit circle (allows me to master one part of trigonometry). Overall this objective is one of the most important objectives for students to comprehend and thus more exciting to explain. = Explanation: =

Step 1) What Vocabulary Do I Need to Know?

 * 1) Sine of an angle= Opposite/Hypotenuse or Y/R
 * 2) Cosine of an angle= Adjacent/Hypotenuse or X/R
 * 3) Tangent of an angle= Opposite/Adjacent or Y/X
 * 4) Unit Circle- the circle whose center is on the origin and whose radius is one unit
 * Equation of Unit Circle x^2 + Y^2=1
 * 1) Know properties for special right triangles.





**Observations:**
>
 * As you can see the radius of the circle begins at the origin of the circle or (0,0).
 * The length of the radius is 1 unit.
 * For the unit circle applying to a coordinate plane:
 * cosine=x
 * sine=y
 * tangent= r

**Step 3) Establishing the Trigonometric ratios for the points of the Unit Circle on the X and Y axis:**
The angles of these points are 0 degrees(x axis), 90 degrees (y axis), 180 degrees(x axis), 270 degrees(y axis), and 360(x axis) degrees.



Observations:
What is the sine, cosine, and tangent values for the ordered pair (1,0)
 * When giving a ordered pair for the unit circle it is (x,y) or (cosine, sine)
 * We can establish the tangent of an angle through taking the sine and dividing it by the cosine
 * Determining Your Level of Understanding So Far:**
 * Question:**
 * Answer:**
 * Sine=0 **
 * Cosine=1 **
 * Tangent=0/1 or 0 **

**Step 4) Establishing 6** **Trigonometric ****Ratios for the Special angle of 30, 45, and 90:**
In order to determine the 6 trigonometric ratios for the angles 30, 45, and 90 it is vital to understand the properties of 45-45-90 triangles and 30-60-90 triangles.

In order to find the ordered pair or trigonometric ratios one must use the properties of a 30-60-90 triangle. Finding x: Its across from what would be 60 degrees and thus the middle length side. To find this side you multiply the shortest side (1/2) by radical 3 which = ** radical 3/ 2. ** Finding y: Its across from the 30 degrees and thus the shortest side. To find this side you have to divide the hypotenuse by 2 giving you **1/2.** **Ordered Pair: (radical 3/2, 1/2)**
 * Trigonometric Ratios for 30 degrees or in radians (Pi)/6:**


 * Trigonometric Ratios for 45 degrees or in radians (Pi)/2:**



In order to find the ordered pair or trigonometric ratios one must use the properties of a 45-45-90 triangle. Finding x and y: Its across from what would be 45 degrees. To find this side you divide the hypotenuse by radical 2. And the x and y end up being 1/ radical 2. However you don't want to leave a radical in the denominator and so you mulitply (1/radical 2) by (radical 2/ radical 2)= ** radical 2/ 2. ** **Ordered Pair: (radical 2/2, radical 2/2)**


 * Trigonometric Ratios for 60 degrees or in radians (Pi)/3:**



In order to find the ordered pair or trigonometric ratios one must use the properties of a 30-60-90 triangle.

Finding x: Its across from what would be 30 degrees and thus the shortest side. To find this side you have to divide the hypotenuse by 2 giving you **1/2.** Finding y: Its across from what would be 60 degrees and thus the middle length side. To find this side you multiply the shortest side (1/2) by radical 3 which gives you = ** radical 3/ 2. **

**Ordered Pair: (1/2, radical 3/2)**

**Recap:**



 * ======So the possible trigonometric ratios are 1/2, radical 2/2, or radical 3/2======

**Step 5) Applying what we Have Learned to Multiples of 30, 45, and 90 Angles**
Now that we have found the possible ratios all we have to do is apply positive or negative signs to the original ordered pairs that we came up with, for the multiples (just like you do for any ordered pair when it is reflected over the x and y axis).

**Step 6) Examples**
1) Find the cosine of 60 degrees. Correct Answer: 1/2

2) When is the cosine radical3/2? Correct Answer: At 30 degrees and 300 degrees. Why? There are two answers because in Quadrant 1 and 4 the x value of radical 3/2 is positive.

3) Find the Tangent of 5(pi)/6. Correct Answer: -radical 3/3 Why? First you figure out what 5(pi)/6 is, which is 150 degrees. Then you find the ordered pair for 150 degrees (-radical3/2, 1/2). Finally, you have to take (1/2) and divide it by -radical3/2= -radical3/3

Step 7) Portfolio Problems
A) What is the ordered pair for 300 degrees? B) What is the ordered pair for 7(pi)/6 C) Find the tangent of 3(pi)/2 D) What is the sine of 3(pi)/4 E) What is the cosine of 240 degrees?

__Instructional Video__
[]

Sources and Pictures: [] [] [] []