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Introduction to the Unit Circle and Right Triangle Trigonometry Presented by, Ginny Hayes Space Coast Jr/Sr High Draw the circle x y 1 2 2 Label the x- and y-intercepts. Your circle should look like this: (0,1) (1,0) (-1,0) (0,-1) Tell me what you know about this circle. (0,1) (1,0) (-1,0) (0,-1) Typical responses include: • • • • • • • It’s round. It has no corners. It has a diameter. It has a radius. It has 360 . 2 r . The area is The circumference is 2r. 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 180 . • A quarter of the way around is a right angle or 90 . • Three-fourths of the way around the circle is 270 . (0,1) 90 . 0 (1,0) (-1,0) 360 180 . (0,-1) 270 . We can further divide up our circle into smaller sections. 45 If we divide the first quadrant in half, our angle is We can repeat this for each of the remaining quadrants. (0,1) 90 . 135 45 0 (-1,0) 360 180 . 225 315 (0,-1) 270 . (1,0) 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) 90 . 120 30 and 60 . 60 150 30 30 0 (-1,0) 180 . 360 210 330 300 60 240 (0,-1) 270. (1,0) Joining the quarters and thirds would give us the following circle: (0,1) 90 . 120 135 60 45 150 30 0 (-1,0) 180 . 360 210 225 330 315 240 (0,-1) 270 . 300 (1,0) 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. 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 • c = a x 2, so a 2 • b = a x 3 , so c 3 b 3 c 2 2 •In a 45-45-90 triangle with hypotenuse “c” and legs “a”: • c = a x 2 , so c a 2 c x, y 2 Using the special triangle relationship with t= 45 and c = 1: 1 So, cos 45 2 1 And, sin 45 2 2 x 2 2 y 2 (0,1) Because the angles are equal, x and y are equal, so the sin ratio will be the same as the cos. 2 2 2 , 2 Y t (-1,0) (1,0) X (0,-1) hyp For the 30 central angle triangle, the shorter leg (y) 2 12 . The longer leg (x) 3 2 hyp 3 2 1 3 2 . 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. x cos 30 y sin 30 x cos 60 y sin 60 3 2 1 2 1 2 3 2 (0,1) 90 . 3 2 1 y sin( 30 ) 2 x cos(30 ) (-1,0) 180 . 1 3 , 2 2 60 30 3 1 2 ,2 0 360 (0,-1) 270. (1,0) Terminal Point Coordinates t cos t x sin t y 0 30 45 1 0 3 2 1 2 60 90 2 2 1 2 0 2 2 3 2 1 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). The coordinates for each terminal point are as follows: 1 3 , 2 2 2 2 2 , 2 (0,1) 90 . 120 1 3 , 2 2 60 2 2 45 135 150 1 2 30 3 2 ,2 2 , 3 1 2 ,2 0 (-1,0) 180 . 3 1 2 , 2 360 3 1 , 2 2 330 210 2 225 2 2 , 2 1 3 , 2 2 (1,0) 315 240 (0,-1) 270 . 2 2 2 , 2 1 3 , 2 2 300 hypotenuse opposite t adjacent • 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!!!! opp sin( t ) hyp adj cos(t ) hyp opp tan( t ) adj 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. 6 1 sin 30 c sin 30 6 c 6 c 12 c 2 B c a=6 3 b b bb6 3 cos 30 cos 30 12 c 12 2 B 90 30 60 C b A 30 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 On your post-it note, write •Pointer: 60 and •Middle: 45 place it •Ring: 30 on your •Pinky: 0 palm. 2 Your hand should look like this: 90 60 45 2 30 0 Here is how it works. Example: Find cos 60 . 1. Fold down the finger with 60 on it. 2. Count the number of fingers above the folded one. 3. Put this number inside the radical on your post-it. 4. This is the value of cos 60 . You should have gotten 1 2 . To find the sin 60 , simply count the fingers below the folded one and place the number in the radical. The value is 3 2 Now for the fancy stuff. What if you wanted to know tan Knowing that tan(t)= opp y sin( t ) adj x cos(t ) 60 ? , place your sin 60 answer over your cos 60 answer and you will get, 3 2 3 1 1 2 3 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! 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 2 and count the fingers above the folded one. 2 For csc, use the reciprocal of sin or fingers below the folded one. and count the 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. When you get comfortable with it, you can use the hand trick backwards when solving trigonometric equations. Example: sin x 2 sin x 3 0 2 sin x 3sin x 1 0 sin x 3 0 sin x 1 0 sin x 3 sin x 1 Can you figure out on your hand how to get an answer of the whole number 3? Or, the whole number 1? 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 24 22 1 . The value of x where sin equals 1 is 90 . 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. Summary of Hand Trick # fingers above folded cos( x) 2 sec( x) 2 # fingers above folded sin( x) # fingers below folded 2 csc( x) 2 # fingers below folded tan( x) # fingers below folded cot( x) # fingers above folded # fingers above folded # fingers below folded 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: [email protected]. fl.us • For a copy of the presentation, send a CD through the courier.