1. Warm Up

2. Reference Angles If you know the reference angle, use these formulas to find the other quadrant angles that have the same reference angle Degrees Radians

3. 8-1 Simple Trigonometric Equations Objective: To Solve Simple Trigonometric Equations and Apply Them

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 x y 1 -19 π 6 -11 π 6 -7 π 6 π 6 5 π 6 13 π 6 17 π 6 25 π 6 -π-π -2 π -3 π π 2π2π 3π3π 4π4π All the solutions for x can be expressed in the form of a general solution. y = y=sin x

6. Solving for angles that are not on UC We will work through solutions algebraically and graphically. Learning both methods will enhance your understanding of the work.

7. Method 1: Algebraically: Step 1 Set the calculator in degree mode and use the inverse sine key

8. Find the final answer(s) for the given range. Since the answer given by your calculator is NOT between 0 and 360 degrees, find the proper answers by using RA.

10. Method 2: Solving Graphically

12. Method 1: Algebraically: Step 1: Set the calculator in radian mode and use the inverse sine key

13. Step 2: Determine the Proper Quadrant

15. Method 2: Graphically Step 1 Set the calculator in radian mode.

16. Use your knowledge of trig functions to choose an appropriate window Use the intersect Key once more for the second point of intersection. i.e solution.

17. When you use the graphing method, you can easily see there is more than one solution. When using the graphing method, it might take a while to set the window properly. The algebraic method is quicker, however, you have to make sure to look for a possible second answer.

18. Example 3

19. Find the appropriate quadrant

20. Another way: ignore the negative sign.

21. Graphing Calculator: Although this is a reasonable window to start with, it does not capture the graph. So change Ymin and Ymax.

23. Inclination and Slope

24. Theorem

25. Example 5

26. 65.2 ˚, 294.8 ˚ 4.1 ˚, 175.9 ˚ -0.8391 31 ˚ 0.83 ˚, 2.31 ˚

28. Homework: Page 299 #1-35 odds Challenge: #45 & 46

29. Set Notation Not Included The interval does NOT include the endpoint(s) Interval NotationInequality NotationGraph Parentheses ( ) < Less than > Greater than Open Dot Included The interval does include the endpoint(s) Interval NotationInequality NotationGraph Square Bracket [ ] ≤ Less than ≥ Greater than Closed Dot

30. Graphically and algebraically represent the following: All real numbers greater than 11 Graph: Inequality: Interval: Example 1 10 11 12 Infinity never ends. Thus we always use parentheses to indicate there is no endpoint.

31. Describe, graphically, and algebraically represent the following: Description: Graph: Interval: Example 2 1 3 5 All real numbers greater than or equal to 1 and less than 5

32. Describe and algebraically represent the following: Describe: Inequality: Symbolic: Example 3 -2 1 4 All real numbers less than -2 or greater than 4 The union or combination of the two sets.

33. Example 4 Domain: Range: Domain: Range: Describe the domain and range of both functions in interval notation:

34. Example 5 Find the domain and range of. t -32-20-155-40123 h 1087-7421ER DOMAIN:RANGE: The range is clear from the graph and table. The input to a square root function must be greater than or equal to 0 Dividing by a negative switches the sign

35. 2 < x < 6 is a compound inequality, read as “2 is less than x, and x is less than or equal to 6.” Compound Inequalities 231456789 A compound inequality consists of two inequalities connected by the words and or or. For example:

36. Compound Inequalities – two possible cases: And/Or 0-2123456 Graph all real numbers that are greater than zero and less than or equal to 4. 0 < x < 4 AND cases have the variable in-between two numbers. The graph is therefore in-between two numbers. This is the INTERSECTION of the Individual Solutions.

37. Compound Inequalities – two cases: And/Or Graph all real numbers that are less than –1 or greater than 2. x 2 0-2123456 OR cases have TWO separate answers and are solved (and graphed) separately (but on the same number line). The graph of this case goes in opposite directions. (This is the Union of the solutions to either inequality)