2. Do Now: given the equation of a circle x 2 + y 2 = 1. Write the center and radius. Aim: What is the unit circle? HW: p.366 # 4,6,8,10,18,20 p.367 # 2,4,6,8

3. A circle with center at (0, 0) and radius 1 is called a unit circle. (0,1) (0,-1) (-1,0) The points on the circle must satisfy x 2 + y 2 = 1 (1,0)

4. Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value? (1,0) (0,1) (0,-1) (-1,0) x = 1/2 You can see there are two y values. They can be found by putting 1/2 into the equation for x and solving for y. We'll look at a larger version of this and make a right triangle.

5. (1,0) (0,1) (0,-1) (-1,0)  We know all of the sides of this triangle. The bottom leg is just the x value of the point, the other leg is just the y value and the hypotenuse is always 1 because it is a radius of the circle. Notice the sine is just the y value of the unit circle point and the cosine is just the x value.

6. (1,0) (0,1) (0,-1) (-1,0) We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions.  So if I want a trig function for  whose terminal side contains a point on the unit circle, the y value is the sine, the x value is the cosine and y/x is the tangent.

7. Unit Circle (1, 0) (0, 1) (-1, 0) (0, -1) 0

8. Quadrantal Angles Definition. A quadrantile angle is an angle whose initial side lies on one of the coordinates axes. Examples. How do we find trig values in this case?

9. Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division? 45° We can label this all the way around with how many degrees an angle would be and the point on the unit circle that corresponds with the terminal side of the angle. We could then find any of the trig functions. 45° 90° 0°0° 135° 180° 225° 270° 315° These are easy to memorize since they all have the same value with different signs depending on the quadrant.

10. Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division? 30° We can again label the points on the circle and the sine is the y value, the cosine is the x value and the tangent is y over x. 30° 90° 0°0° 120° 180° 210° 270° 330° You'll need to memorize these too but you can see the pattern. 60° 150° 240° 300°

11. You should memorize this. This is a great reference because you can figure out the trig functions of all these angles quickly.

12. Look at the unit circle and determine sin 420°. All the way around is 360° so we’ll need more than that. We see that it will be the same as sin 60° since they are coterminal angles. So sin 420° = sin 60°. In fact sin 780° = sin 60° since that is just another 360° beyond 420°. All of these would have a sine value that is the y value of the point on the unit circle here.

13. Now let’s look at the unit circle to compare trig functions of positive vs. negative angles. Remember negative angle means to go clockwise

16. You can memorize the unit circle easily because you know the points on the axis are 0’s and 1’s with negatives in appropriate places. All of the multiples of 45  that don’t land on an axis have the value for both x and y with the appropriate sign for the quadrant. All of the multiples of 30  have the values with appropriate signs for the quadrant. You can tell which one of these comes first because is bigger than so when x is bigger put it first.

17. For angles that are multiples of 30  they only have one of 2 possible values for x and y if they don’t land on an axis. If the x is further you know that one is the square root of 3 over 2, if it is not as far over as the y is up or down you know it is ½. 30° 90° 0°0° 120° 180° 210° 270° 330° 60° 150° 240° 300°