4.2: The Unit Circle Objectives: To construct all aspects of the unit circle (angles and points) To use the unit circle to find the values of trig functions.

Exercise 2 The hypotenuse of a 45-45-90 right triangle has length 1 unit. Find the length of the other two sides of the triangle. Add this to your worksheet.

Exercise 3 The hypotenuse of a 30-60-90 right triangle has length 1 unit. Find the length of the other two sides of the triangle. Add this to your worksheet.

Unit Circle: 0-0-0 Now let’s build our unit circle one triangle at a time. Except the first set of points do not really make a triangle. Oh, well. I think at some point we said the radius of this circle is 1. Which four points does this nugget of information help us find?

Unit Circle: 45-45-90 Next, let’s find the coordinates of the points that create angles whose measures are multiples of 45°. Well, 45°, 135°, 225°, and 315° anyway.

Unit Circle: 45-45-90 To get to the point in quadrant 1 rotated 45°, we could do one of two things. We could rotate 45° and then travel 1 unit. That’s called polar coordinates.

Unit Circle: 45-45-90 The second thing we could do is travel to the right a bit and then up a bit. This forms a 45-45-90 right triangle. What are the leg lengths? What are the coordinates of the point?

Unit Circle: 45-45-90 The next point could be found by rotating 135° and then traveling 1 unit. Alternatively, we could use rectangular coordinates by going left a bit and up a bit. Another 45-45-90 right triangle.

Unit Circle: 45-45-90 One thing to bear in mind for these coordinates is that since we had to go left, our x-coordinate is negative.

Unit Circle: 45-45-90 Next, let’s rotate 225° and travel 1 unit. Or maybe we want to go left a bit and then down a bit. Watch your signs.

Unit Circle: 45-45-90 Finally, we’ll rotate 315° and travel 1 unit. Which is the same as going right a bit and then down a bit. Watch your signs again.

Exercise 4 Use a special right triangle to find the exact and approximate value of each of the following. sin 45° cos 45° tan 45° sin π/4 cos π/4 tan π/4

Exercise 5 Use a calculator to find the approximate value of each of the following. sin 135° cos 135° sin 225° cos 225° sin 315° cos 315° sin 3π/4 cos 3π/4 sin 5π/4 cos 5π/4 sin 7π/4 cos 7π/4

Unit Circle: 30-60-90 Now that we’ve finished with the 45-45-90 triangles, let’s find the points at 30°, 150°, 210°, and 330°

Unit Circle: 30-60-90 For that first point, we could rotate 30° and the travel 1 unit, using polar coordinates, or we could use good old-fashioned rectangular coordinates by moving right some and then up a bit less.

Unit Circle: 30-60-90 Using rectangular coordinates, of course, creates another right triangle, a 30-60-90. What are the lengths of the legs? What are coordinates of the point?

Unit Circle: 30-60-90 Now we’ll rotate 150° and then travel 1 unit. Which is really the same as moving left some and up a bit less. What do you know, another 30-60-90 right triangle.

Unit Circle: 30-60-90 Next we’ll rotate 210° and then travel 1 unit. Or maybe we should move left some and down a bit less. Watch your signs.

Unit Circle: 30-60-90 Perhaps you’re beginning to get the picture. Let’s do one more point here, rotated at 330°. Keep an eye on those signs.

Exercise 6 Use a special right triangle to find the exact and approximate value of each of the following. sin 30° cos 30° tan 30° sin π/6 cos π/6 tan π/6

Exercise 7 Use a calculator to find the approximate value of each of the following. sin 150° cos 150° sin 210° cos 210° sin 330° cos 330° sin 5π/6 cos 5π/6 sin 7π/6 cos 7π/6 sin 11π/6 cos 11π/6

Unit Circle: 60-30-90 Finally, let’s move on to a completely different triangle, the 60-30-90 right triangle so we can get the coordinates of the points at 60°, 120°, 240°, and 300°.

Exercise 8 Use a special right triangle to find the exact and approximate value of each of the following. sin 60° cos 60° tan 60° sin π/3 cos π/3 tan π/3

Exercise 9 Use a calculator to find the approximate value of each of the following. sin 120° cos 120° sin 240° cos 240° sin 300° cos 300° sin 2π/3 cos 2π/3 sin 4π/3 cos 4π/3 sin 5π/3 cos 5π/3

A Very Useful Unit Circle Let’s recap: We know all the angle measures on the unit circle in both degrees and radians, and we know the coordinates of the points along the unit circle. Maybe this would be a bit more useful if we put it all together on one circle.

Unit Circle: Trig Functions For a unit circle, let t be an angle in standard position whose terminal side intersects the point (x, y) on the circle.

Domain: Sine and Cosine Recall that the domain of a function is the set of inputs. For trig functions, this is the set of angle values that we are allowed to evaluate. For both sine and cosine, the domain is all real numbers. In other words, you can evaluate the sine or cosine of an angle, positive or negative, even angles over 360°.

Range: Sine and Cosine Recall that the range of a function is the set of outputs. On the unit circle, that’s the set of all x-coordinates for cosine and the set of all y-coordinates for sine. Since the radius of the unit circle is 1, this means that sine and cosine will always be between -1 and 1.

Master Trig Table Now we can use the unit circle to find trig values of angles from 0° to 360° or 0 to 2π radians. There’s at least a hundred of the things. Let’s organize it all in a handsome table.

Evaluate the six trig functions at t = 2π/3 Exercise 10 Evaluate the six trig functions at t = 2π/3

Evens and Odds Cosine and Secant are even: Sine, Cosecant, Tangent, and Cotangent are odd:

Evaluate the six trig functions at t = 13π/6. Exercise 11 Evaluate the six trig functions at t = 13π/6.

Periodic Functions Of course we can measure our angles over 2π radians. Likewise, we can evaluate trig functions at these angles; it’s just that they start over after 2π. Then they repeat again after 4π. And again after 6π. Functions that have this cyclical behavior are called periodic functions.

Periodic Functions A function f is periodic if there exists a positive real number c such that For all t in the domain of f. The smallest number c for which f is periodic is called the period of f.

Periodic Functions For sine and cosine, the period is 2π. This means that all of the values for sine and cosine repeat after multiples of 2π.

4.2: The Unit Circle Objectives: To construct all aspects of the unit circle (angles and points) To use the unit circle to find the values of trig functions. Assignment: P. 299: 1-4 S P. 299: 13-22 S P. 299: 23-28 S P. 299: 29-36 S P. 300: 50, 60 Memory quiz next class (30 Q in 30 min)