1. Assignment P. 485-489: 2-8, 10- 28 even, 30, 31, 34, 42 Complete Unit Circle Challenge Problems

2. Example 1 What if you knew two sides of a right triangle, how could you find the measure of an angle opposite one of those sides? In other words, if the legs of a right triangle are 3 and 4, what is the measure of the angle opposite the smallest side?

3. Example 2 What does an inverse do to a function algebraically and graphically? Inverses switch inputs and outputsInverses reflect a graph over y = x Inverses give you a way to find the input when you know the output.

4. Example 2 Explain what x and y represent in y = sin x, then explain what is meant by the inverse of y = sin x. The inverse would switch these: Input = ratio of sides Output = angle measure

5. Example 3 For what value of x is ( x, 3) on the graph of f(x) = 2 x – 5?

6. Example 4 For what value of x is ( x, 4) on the graph of g(x) = x 2 – 5?

7. One-to-One Functions one-to-one The difference between the two previous examples is that the linear function is one-to-one but the quadratic is not. In other words, for the linear function every input has one output, AND every output has one input. For functions that are not one-to-one, a given output may have multiple inputs.

8. Example 5 Use the graph of f ( x ) = sin( x ) to explain why a given output may have different inputs.

9. Trigonometric Ratios II Objectives: 1.To find missing angles in a right triangle using inverse trigonometric ratios 2.To complete and use the unit circle to find the exact values of various angle measures

10. Inverse Trigonometry To find an angle measurement in a right triangle given any two sides, use the inverse of the trig ratio, but each of them are only defined on certain intervals.

11. Inverse Trigonometry

12. “sin -1 x ” is read “the angle whose sine is x ” or “inverse sine of x ” arcsin x is the same thing as sin -1 x

13. Example 5 Let <A and <B be acute angles in a right triangle. Use a calculator to approximate the measures of <A and <B to the nearest tenth of a degree. 1.sin A = 0.87 2.cos B = 0.15 Type in your calculator as:

14. Example 6 If the legs of a right triangle are 3 and 4, what is the measure of the angle opposite the smallest side?

15. Example 7 Find the measures of the acute angles of a 8-15-17 right triangle.

16. Example 8 Suppose your school is building a raked stage. The stage will be 30 feet long from front to back, with a total rise of 2 feet. A rake (angle of elevation) of 5° or less is generally preferred for the safety and comfort of the actors. Is the raked stage you are building within this suggested range?

18. Example 9 solve a right triangle To solve a right triangle means to find all of its sides and angles. Using trigonometry, what must you know to solve a right triangle?

19. Example 10 Solve the right triangle. Round your answers to the nearest tenth.

20. Example 11 Solve each right triangle. Write your answers in simplest radical form.

21. Radians

23. Radians for the Smart Masses

24. Radians radian Radians are another way to measure an angle. If you take the radius and wrap it around the circle, the angle that is formed is one radian.

25. Radians It takes a little bit more than 3 radians to span a semicircle. That “little bit more than 3” is π. So π radians = 180° and 2 π radians = 360°

26. Example 10 Rewrite each of the following angle measures in terms of radians. (180° = π rad) 1.30° 2.45° 3.60° 4.90°

27. The Unit Circle unit circle This tiny circle is called the unit circle since its radius is 1 unit. This circle may be tiny, but it will give us a way to find 102 exact trig values. That’s pretty useful.

28. Unit Circle Activity unit circle Math students often use a unit circle to find the exact trig ratios of certain angle measures, since most calculators won’t divulge that information. In this activity, we will construct a unit circle.

29. Unit Circle Activity The outer bold circle is the unit circle. We will eventually be labeling each of the points along this bold circle with ordered pairs.

30. Unit Circle Activity Now look at the inner circle. The points along this circle will be labeled with degree measures. What do you suppose these degree measures represent? Finish the degree measures.

31. Unit Circle Activity Now look at the middle circle. The points along this circle will be labeled with radian measures. Finish the radian measures for the first quadrant.

32. Unit Circle Activity Finally, look at the outer circle again. Let’s concentrate on the point that is 30° along this circle.

33. Unit Circle Activity Finally, look at the outer circle again. Let’s concentrate on the point that is 30° along this circle. Realize that the ordered pair for any point makes a right triangle with the x -axis.

34. Unit Circle Activity 1.What is the length of the hypotenuse? 2.What is the length of the short leg? 3.What is the length of the longer leg? 4.What are the coordinates of the point on the circle?

35. Unit Circle Activity How can this unit circle be used to find the following? 1.cos (30°) 2.sin (30°) 3.tan (30°)

36. Unit Circle Activity Let’s look at the outer circle again. This time concentrate on the point that is 45° along this circle.

37. Unit Circle Activity 1.What is the length of the hypotenuse? 2.What is the length of the base leg? 3.What is the length of the height leg? 4.What are the coordinates of the point on the circle?

38. Unit Circle Activity How can this unit circle be used to find the following? 1.cos (45°) 2.sin (45°) 3.tan (45°)

39. Unit Circle Activity In general, for any point ( x, y ) along the outer circle of the unit circle: 1.cos(  ) = x 2.sin (  ) = y 3.tan (  ) = y/x

40. Unit Circle Activity Let’s look at the outer circle one last time. This time let’s look at the point that is 150° along the unit circle. Obviously, we cannot make a right triangle with a 150° angle. So how could we complete the second quadrant?

41. Unit Circle Activity The answer involves symmetry.

42. Unit Circle Activity The same would apply for the 3 rd and 4 th quadrants.

43. Assignment P. 485-489: 2-8, 10- 28 even, 30, 31, 34, 42 Complete Unit Circle Challenge Problems