LESSON 5 Section 6.3

1. LESSON 5 Section 6.3 Trig Functions of Real Numbers

2. UNIT CIRCLE Remember, the sine of a real number t (a number that corresponds to radians) is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle. APPENDIX IV of your textbook shows a good unit circle.

3. Make a table of x and y values for the equation y = sin x.

4. This is a second revolution around the unit circle. This is another ‘period’ of the curve.

5. y = sin x • This is a periodic function. The period is 2π. • The domain of the function is all real numbers. • The range of the function is [-1, 1]. • It is a continuous function. The graph is shown on the next slide.

6. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Graphing the sine curve for -2π ≤ x ≤ 2π. (π/2, 1) (2π, 0) (π, 0) (0, 0) (3π/2, - 1)

7. UNIT CIRCLE Remember, the sine of a real number t (a number that corresponds to radians) is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle.

8. Make a table of x and y values for y = cos x Remember, the y value in this table is actually the x value on the unit circle.

9. y = cos x • This is a periodic function. The period is 2π. • The domain of the function is all real numbers. • The range of the function is [-1, 1]. • It is a continuous function. The graph is shown on the next slide.

10. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Graphing the cosine curve for -2π ≤ x ≤ 2π. (0, 1) (2π, 1) (π/2, 0) (3π/2, 0) (π, - 1)

11. How do the graphs of the sine function and the cosine function compare? • They are basically the same ‘shape’. • They have the same domain and range. • They have the same period. • If you begin at –π/2 on the cosine curve, you have the sine curve.

12. The notation above is interpreted as: ‘as x approaches the number π/6 from the right (from values of x larger than π/6), what function value is sin x approaching?’ Since the sine curve is continuous (no breaks or jumps), the answer will be equal to exactly the sin (π/6) or ½ . The notation below is interpreted as: ‘as x approaches the number π/6 from the left (from values of x smaller than π/6), what function value is sin x approaching?’ Again, since the sine curve is continuous, the answer will be equal to exactly the sin (π/6) or ½ .

13. Answer the following.

14. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation. Use the graph to verify these values.

15. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation. Q I Q IV

16. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation.

17. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation.Use the graph to verify these values.

18. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation. Q IV Q III

19. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation.

20. Make a table of x and y values for y = tan x Remember, tan x is (sinx / cosx).

21. y = tanx • This is a periodic function. The period is π. • The domain of the function is all real numbers, except those of the form π/2 +nπ. • The range of the function is all real numbers. • It is not a continuous function. The function is undefined at -3π/2, -π/2, π/2, 3π/2, etc. There are vertical asymptotes at these values. The graph is shown on the next slide.

22. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Graphing the tangent curve for -2π ≤ x ≤ 2π. (π/4, 1) (-π/4, -1)

23. For all x values where the tangent curve is continuous, approaching from the left or the right will equal the value of the tangent at x. However, the two cases above are different; because there is a vertical asymptote when x = -π/2. If approaching from the left (the smaller side), the answer is infinity. If approaching from the right (the larger side), the answer is negative infinity.

24. Find the answers.

25. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation. tan x = 1 Q I Q III

26. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval [0, 2) that satisfy the equation.

27. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation.

28. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation. Q I Q III

29. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation.

30. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation.

31. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation. Q IV Q II

32. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Find all the values x in the interval that satisfy the equation.

33. Sketch the graph of y = sin x + 1 This will be a graph of the basic sine function, but shifted one unit up. The domain will be all real numbers. What would be the range? Since the range of a basic sine function is [-1, 1], the domain of the function above would be [0, 2].

34. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Sketch the graph of y = sin x + 1

35. Sketch the graph of y = cos x - 2 This would be the graph of a basic cosine function shifted 2 units down. The domain is still all real numbers. What is the range? The basic cosine function has a range of [-1, 1]. The range of the function above would be [-3, -1].

36. π -π -π 2 π 2 -3π 2 2π 3π 2 -2π Sketch the graph of y = cos x - 2

37. Find the intervals from –2π to 2πwhere the graph of y = tan x is: • Increasing • Decreasing • Remember: No brackets should be used on values of x where the function is not defined. • Increasing: [-2π, -3π/2) b) The function never decreases. • (-3π/2, -π/2) • (-π/2, π/2) • (π/2, 3π/2) • (3π/2, 2π]