1.6 Trigonometric Functions: The Unit circle

The Unit Circle A circle with radius of 1 Equation x2 + y2 = 1

Do you remember 30º, 60º, 90º triangles?

Do you remember 45º, 45º, 90º triangles?

Do you remember 45º, 45º, 90º triangles? When the hypotenuse is 1 The legs are

Let's pick a point on the circle 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)

We know all of the sides of this triangle 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.

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.

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? 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.

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.

You'll need to memorize these too but you can see the pattern. 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.

We'll see them all put together on the unit circle on the next screen. 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.

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

The Unit Circle with Radian Measures

The Six Trig functions Reciprocal Identities

Lets find the six trig functions if Think where this angle is on the unit circle.

Find the six trig functions of Think where this angle is on the unit circle.

How about

There are times when Tan or Cot does not exist. At what angles would this happen?

Ex 1: Find the values of the sine and cosine functions of an angle in standard position with measure θ if the point (3,4) lies on it’s terminal side.   Ex 2: If the point (5,12) lies on its terminal side.

Ex 3: Find the sin θ when cos θ = and the terminal side of θ is in the 1st quadrant.   Ex 4: Find the sin θ when cos θ = and the terminal side of θ is in the 1st quadrant.

7.) The terminal side of an angle θ in standard position contains the point with coordinates (8,-15). Find the value of all six trig functions.   8.) contains the point (-3,-4)

9.) If csc θ = -2 and θ lies in Quad III, find the values of the five trig functions.   10.) If sec θ = 2 and θ lies in Quad IV:

YOUR TURN!!! FILL IN A BLANK UNIT CIRCLE!! YAAAHHHHH!!!