1 / 17

THE UNIT CIRCLE

THE UNIT CIRCLE. A circle with center at (0, 0) and radius 1 is called a unit circle. Remember from College Algebra that the equation of this circle would be. (0,1). (-1,0). (1,0). (0,-1). So points on this circle must satisfy this equation.

willow-massey
Download Presentation

THE UNIT CIRCLE

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. THE UNIT CIRCLE

  2. A circle with center at (0, 0) and radius 1 is called a unit circle. Remember from College Algebra that the equation of this circle would be (0,1) (-1,0) (1,0) (0,-1) So points on this circle must satisfy this equation.

  3. 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? 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. x = 1/2 (0,1) (-1,0) (1,0) We'll look at a larger version of this and make a right triangle. (0,-1)

  4. 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. (0,1) (-1,0) (1,0)  (0,-1) Notice the sine is just the y value of the unit circle point and the cosine is just the x value.

  5. 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. (0,1) (-1,0) (1,0)  (0,-1) We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions.

  6. Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division? These are easy to memorize since they all have the same value with different signs depending on the quadrant. 90° 135° 45° 180° 45° 0° 225° 315° 270° 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.

  7. Can you figure out what these angles would be in radians? 90° 135° 45° 180° 0° 225° 315° 270° The circle is 2 all the way around so half way is . The upper half is divided into 4 pieces so each piece is /4.

  8. Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division? You'll need to memorize these too but you can see the pattern. 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270° 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.

  9. Can you figure out what the angles would be in radians? We'll see them all put together on the unit circle on the next screen. 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270° It is still  halfway around the circle and the upper half is divided into 6 pieces so each piece is /6.

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

  11. Look at the unit circle and determine sin 420°. 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. 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°.

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

  13. 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.

  14. So if you need the trig value of a multiple of 45 you could draw the circle or where the angle would be on the circle and figure out the x and y values of thepoints there. Then x would be the cosine value and y would be the sign value. Tangent would be yover xand you could take reciprocals for the other values. These are easy to memorize since they all have the same value with different signs depending on the quadrant. 90° 135° 45° 180° 45° 0° 225° 315° 270°

  15. 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. 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270°

More Related