THE UNIT CIRCLE.

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Presentation transcript:

THE UNIT CIRCLE

Remember in the last unit, we talked about a circle with r = 1 unit? (1,0) (0,1) (0,-1) (-1,0) We call this the unit circle because the radius is one unit.

The equation of this circle is Points on this circle must satisfy this equation. (1,0) (0,1) (0,-1) (-1,0)

Let's pick a point on the circle. We'll choose a point where the x is ½. If the x is ½, what is the y value? You can see there are two y values. They can be found by putting ½ into the equation for x and solving for y. x = ½ (0,1) (-1,0) (1,0) (0,-1)

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

So if I want a trig function for  whose terminal side contains a point on the unit circle, the y value is the sine, and the x value is the cosine . (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.

Here is the unit circle divided into 8 pieces Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division? 90° 135° 45° 180° 0° 225° 315° 270°

Can you figure out what these angles would be in radians? 90° 135° 45° 180° 0° 225° 315° 270°

Here is the unit circle divided into 12 pieces Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division? 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270°

Can you figure out what the angles would be in radians? 90° 120° 60° 150° 30° 180° 30° 0° 210° 330° 240° 300° 270°

Homework Memorize the unit circle! You will need to because these angles are commonly used in math! You will be quizzed three times, Tuesday, Wednesday, and Thursday. When you get an A, you will stop quizzing. If you don’t get an A after Thursday, you will keep your highest grade.