Introduction to the Unit Circle and Right Triangle Trigonometry

1. Introduction to the Unit Circle andRight Triangle Trigonometry Borrowed from Ginny Hayes Space Coast Jr/Sr High

2. Draw the circle Label the x- and y-intercepts. Your circle should look like this: (0,1) (1,0) (-1,0) (0,-1)

3. Tell me what you know about this circle. (0,1) (1,0) (-1,0) (0,-1)

4. Typical responses include: • It’s round. • It has no corners. • It has a diameter. • It has a radius. • It has . • The area is • The circumference is

5. Let’s look at the degrees. • Degrees measure angles. What are some angles we can fill in to our circle? • Halfway around the circle is a straight angle or • A quarter of the way around is a right angle or • Three-fourths of the way around the circle is (0,1) (1,0) (-1,0) (0,-1)

6. We can further divide up our circle into smaller sections. If we divide the first quadrant in half, our angle is We can repeat this for each of the remaining quadrants. (0,1) (-1,0) (1,0) (0,-1)

7. I could have divided the 1st quadrant into thirds.If so, the angles would be multiples of 30.This means my circle would look like this: (0,1) (1,0) (-1,0) (0,-1)

8. Joining the quarters and thirds would give us the following circle: (0,1) (1,0) (-1,0) (0,-1)

9. This circle is called the “unit” circle because the radius is 1 unit.Each angle is considered to be in standard position because it starts at 0 degrees and rotates counterclockwise to the terminal point which is where the leg of the angle intersects the unit circle.Our next task is to find the terminal point (x,y) for each angle on the unit circle. We can use properties of symmetry (x-axis, y-axis, and origin) to help us complete this task very quickly.

10. Let’s review our special triangles from geometry. • In a 30-60-90 triangle with hypotenuse “c”, short leg = a and long leg = b: • c = a x 2, so • b = a x , so • In a 45-45-90 triangle with hypotenuse “c” and legs “a”: • c = a x , so

11. Using the special triangle relationship with t= 45 and c = 1: So, cos And, sin (0,1) Because the angles are equal, x and y are equal, so the sin ratio will be the same as the cos. Y t (-1,0) (1,0) X (0,-1)

12. Interchanging the position of the 30 and 60 degree angles will switch the shorter leg to x and the longer leg to y, so the sin and cos values will trade. (0,1) (1,0) (-1,0) (0,-1)

13. Terminal Point Coordinates

14. To complete coordinates in the other quadrants, use symmetry. • In the second quadrant, points are symmetric across the y-axis so the coordinates will be (-x,y). • In the third quadrant, points are symmetric across the origin so the coordinates will be (-x,-y). • In the fourth quadrant, points are symmetric about the x-axis so the points will be (x,-y).

15. The coordinates for each terminal point are as follows: (0,1) (1,0) (-1,0) (0,-1)

16. From geometry, we know sin(t), cos(t), and tan(t). • Sin(t) is the ratio of the opposite side of the triangle to the hypotenuse. • Cos(t) is the ratio of the adjacent side to the hypotenuse. • Tan(t) is the ratio of the opposite side to the adjacent side. • SOHCAHTOA!!!! hypotenuse opposite t adjacent

17. By learning the unit circle coordinate values, a variety of problems can be easily solved without the use of a calculator. For example: Using the information shown, solve the right triangle. B c a=6 C b

18. A “handy” tool for remembering the values of the coordinates for the x or cos values and y or sin values on the unit circle is the hand trick. • Take your labels and write 0, 30, 45, 60, and 90 on them. • Place them on the fingers of your left hand (palm up) • as follows: • Thumb: 90 • Pointer: 60 • Middle: 45 • Ring: 30 • Pinky: 0 On your post-it note, write and place it on your palm.

19. Your hand should look like this:

20. Here is how it works. • Example: Find cos . • Fold down the finger with 60 on it. • Count the number of fingers above the folded one. • Put this number inside the radical on your post-it. • This is the value of cos . • You should have gotten . • To find the sin , simply count the fingers below the folded one and place the number in the radical. The value is

21. Now for the fancy stuff. What if you wanted to know tan ? Knowing that tan(t)= , place your sin answer over your cos answer and you will get, So, you can just put your radical sin number over your radical cos number and you have tan. The 2’s in the denominators will always cancel out!

22. What about sec? Sec is the reciprocal function of cos. Find the cos value and flip it, you now have sec. This means you would use and count the fingers above the folded one. For csc, use the reciprocal of sin or and count the fingers below the folded one. For cot, use the reciprocal of tan or and put the number above the folded one in the top radical and the numbers below the folded one in the bottom radical.

23. When you get comfortable with it, you can use the hand trick backwards when solving trigonometric equations. Example: Can you figure out on your hand how to get an answer of the whole number 3? Or, the whole number 1?

24. There is not a way to get the whole number 3. This means that there is no value of x such that sinx=-3. In order to get an answer of 1, fold down the thumb and you have 4 fingers below the folded one. This would be . The value of x where sin equals 1 is . While this only works for the exact values on the unit circle, it is really a time saver. Students learn the first quadrant, use properties of symmetry, and now they can figure out any exact value problem for any trig function.

25. Summary of Hand Trick

26. My students really enjoyed learning this last year. They found it much easier to remember their exact values. • If you would like copies of the slides, you may e-mail me at: hayesv@brevard.k12.fl.us • For a copy of the presentation, send a CD through the courier.