Fun with the Unit Circle

Putting the pieces together!

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To put the pieces together all you have to do is draw your triangles. For any angle that is a multiple of 30 degrees or the radian equivlant (seen to left) you use your triangle to figure it out. Note due to computer malfunction pi/x will stand for radians.


Easy Example 1

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What is sin of 30 degrees?
1) Draw your appropriate triangle. In this case the 30-60-90. 
2) Find the angle in question in this case the 30 degrees.
3) Sin is opposite/hypotenuse so on the triangle see that opposite of 30 is 1. 
4) Then see that the hypotenuse is 2.
5) plug those numbers into opposite/ hypotenuse and find that the answer is...  

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Easy Example 2

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What is cos of 45 degrees?
1) Draw the appropriate triangle, in this case the 45-45-90 triangle.
2) Find the angle in question. In this case either of the 45 degrees will work.
3) Cos is adjacent/hypotenuse. Find the adjacent side to the degree and see that it is 1.
4) Find the hypotenuse which is (seen below left)   
5) Plug those numbers into adjacent/hypotenuse and you will get (seen below right)
6) Since square roots can not be on the bottom you must simplify by multiplying top and bottom by the square root of 2.   

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Easy Example 3

What is tan of 60 degrees?
Do it yourself!

Medium Example 1

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What is sin of 7pi/6? 
1) Change radians into degrees by multiplying the bottom by 180 and dividing the top. You will find that it equals 210 degrees which is a 30 degree multiple.
2) Draw your appropriate triangle. In this case the 30-60-90. 
3) Find the angle in question in this case the 30 degrees.
4) Sin is opposite/hypotenuse so on the triangle see that opposite of 30 is 1. 
5) Then see that the hypotenuse is 2.
                                               6) Remember that in this quadrant sin is negative!    
                                               7) Plug those numbers into opposite/ hypotenuse and find that the answer is...  


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Medium Example 2

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What is tan of 7pi/4?
1) Change radians into degrees by multiplying the bottom by 180 and dividing by the top. In this case it comes out to 315 degrees.
2) Draw the appropriate triangle, in this case the 45-45-90 triangle.
3) Find the angle in question. In this case either of the 45 degrees will work.
4) Tan is opposite/adjacent. Find the opposite side to the degree and see that it is 1.
5) Find the adjacent side which is also 1.   
                                       6) Plug those numbers into opposite/adjacent and you will get 1
                                       7) Remember that in this quadrant tangent is negative.
                                                                                                   - 1

Medium Example 3

What is cos of 2pi/3?
Do it yourself!

Hard Example 1

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What is csc of -210 degrees?
1) Subtract 210 from 360 to get 150 degrees. 
2) Draw your appropriate triangle. In this case the 30-60-90. 
3) Find the angle in question in this case the equivalent of 30 degrees.
4) Csc is hypotenuse/opposite so on the triangle see that hypotenuse 2. 
5) Then see that the opposite of 30 degrees 1.   
                                              7) Plug those numbers into hypotenuse/opposite and simplify then see that the answer is...
                                                                                                         


Hard Example 2

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What is sec of -pi/4?
1) Change radians into degrees by multiplying the bottom by 180 and dividing by the top. In this case it comes out to 45 degrees.
2)Subtract 45 from 360 to get 315 degrees. 
3) Draw the appropriate triangle, in this case the 45-45-90 triangle.
4) Find the angle in question. In this case either of the 45 degrees will work.
5) Sec is hypotenuse/adjacent. Find the hypotenuse and see that it is square root 2.
                                       6) Find the adjacent side which is 1.   
                                       7) Plug those numbers into adjacent/hypotenuse and see that the answer is... 
                                       

Hard Example 3

What is cot of -240 degrees?
Do it yourself!

Real World Application 1

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Traveling from point a to point b:  from point a you can travel one unit to the east, take a 90 degree turn north and travel another single unit.  if you were to turn 45 degrees north-east and travel in one line, what is that distance? Find the other 6 trig functions that go with the correct triangle.
1) Identify the triangle being used (45-45-90).
2) Draw the picture and label sides.
3) Find the missing side of the 45-45-90 triangle.
4) Answers: (root 2)


Real World Application 2

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Find the height of a telephone pole. You are standing one unit away from the base of a pole. To see the top of the pole you must look up 60 degrees, you know the distance is two units. What is the height of the pole? Find the other 6 trig functions that go with the correct triangle.
1) Identify the triangle in use (30-60-90)
2) Draw and label the triangle
3) Find the missing side of the 30-60-90
4) Answers: (root 3)

Real World Application 3

A bug crawls a distance of (root 3) units, turns 90 degrees and travels one more unit. If the bug were to crawled a straight line between where he started and were he ended, what would the distance be? Find the other 6 trig functions corresponding to the correct triangle.
Do it yourself!