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THE UNIT CIRCLE

THE UNIT CIRCLE. Our goal in todays lesson will be to build the parts of this unit circle. You will then want to get it memorized because you will use many facts from this to answer other Pre Calc and Calculus questions. A circle with center at (0, 0) and radius 1 is called a unit circle.

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THE UNIT CIRCLE

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  1. THE UNIT CIRCLE

  2. Our goal in todays lesson will be to build the parts of this unit circle. You will then want to get it memorized because you will use many facts from this to answer other Pre Calc and Calculus questions

  3. A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be (0,1) (-1,0) (1,0) (0,-1) So points on this circle must satisfy this equation.

  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? 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 can also use special right triangles and trig functions to find important points on the unit circle! (0,-1)

  5. Special right triangle relationship reminders

  6. 3 Trig Functions you learned in Geometry SOH CAH TOA a d j a c e n t θ hypotenuse opposite

  7. 3 More Trig Functions based on reciprocals a d j a c e n t θ hypotenuse opposite We will look closer at these later in the lesson!

  8. Let’s use special right triangles and trig functions to fill in the x-y coordinates of this unit circle! 0 1 Radius = 1 30 : 60 : 90 1 0 x : : 2x -1 0 : : 0 -1

  9. 0 1 P There is a relationship between the coordinates of a point P on the circle and the sine and cosine of the angle (θ) containing P 1 0 -1 0 P ( cos θ, sin θ ) 0 -1

  10. Now lets complete quadrant I for 45° and 60° as well!

  11. Let’s use special right triangles and trig functions to fill in the x-y coordinates of this unit circle! P 0 1 Radius=1 45 : 45 : 90 x : x : 1 0 -1 0 : : 0 -1

  12. Let’s use special right triangles and trig functions to fill in the x-y coordinates of this unit circle! P 0 1 P ( cos θ, sin θ ) 1 0 -1 0 0 -1

  13. Let’s use special right triangles and trig functions to fill in the x-y coordinates of this unit circle! P 0 1 30 : 60 : 90 1 0 -1 0 x : : 2x : : 0 -1 Same result as when we used circle formula!!!

  14. 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° 45° is the reference angle for 135°, 225°, and 315 °

  15. Reference Angles 45° is the reference angle for 135°, 225°, and 315 ° 30° is the reference angle for 150°, 210°, and 330 ° 60° is the reference angle for 120°, 240°, and 300 ° Use the points we found in quadrant I and consider the signs of each quadrant and the reference angles to find the remaining coordinates in the unit circle.

  16. Complete the angles from the reference angle: 30° 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270°

  17. Complete the angles from the reference angle: 60° 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270°

  18. 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.

  19. 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.

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

  21. Look at the unit circle and determine sin 420°. sin 420° = sin 780° = 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°.

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