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. xyxy 007π/6-0.5 π/60.55π/4-0.707 π/40.7074π/3-0.866 π/30.866-3π/2 π/215π/3-0.866 2π/30.8667π/4-0.707 3π/40.70711π/6-0.5 5π/60.52π2π0 π013π/60.5

4. xyxy 2π2π019π/6-0.5 13π/60.513π/4-0.707 9π/40.70710π/3-0.866 7π/30.8667π/2 5π/2111π/3-0.866 8π/30.86615π/4-0.707 11π/40.70723π/6-0.5 17π/60.54π4π0 3π3π025π/60.5 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. Graphing the sine curve for -2π ≤ x ≤ 2π. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π (0, 0) (π/2, 1) (π, 0) (3π/2, - 1) (2π, 0)

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 xyxy 017π/6-0.866 π/60.8665π/4-0.707 π/40.7074π/3-0.5 π/30.5-3π/20 π/205π/30.5 2π/3-0.57π/40.707 3π/4-0.70711π/60.866 5π/6-0.8662π2π1 π13π/60.866 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. Graphing the cosine curve for -2π ≤ x ≤ 2π. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π (0, 1) (π/2, 0) (π, - 1) (3π/2, 0) (2π, 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. Find all the values x in the interval [0, 2  ) that satisfy the equation. Use the graph to verify these values. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

15. Find all the values x in the interval [0, 2  ) that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π Q IQ IV

16. Find all the values x in the interval [0, 2  ) that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

17. Find all the values x in the interval [0, 2  ) that satisfy the equation. Use the graph to verify these values. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

18. Find all the values x in the interval [0, 2  ) that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π Q III Q IV

19. Find all the values x in the interval [0, 2  ) that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

20. Make a table of x and y values for y = tan x xyxy -π/2undefined0.49π31.821 -0.49π-31.821π/2undefined -π/3-1.7320.51π-31.821 -π/42π/3-1.732 -π/6-0.5773π/4 005π/6-0.577 π/60.577π0 π/417π/60.577 π/31.7325π/41 Remember, tan x is (sinx / cosx).

21. y = tan x 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. Graphing the tangent curve for -2π ≤ x ≤ 2π. -π 2 -π -3π 2 -2π π2π2 π 3π 2 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. Find all the values x in the interval [0, 2  ) that satisfy the equation. tan x = 1 -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π Q IQ III

26. Find all the values x in the interval [0, 2  ) that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

27. Find all the values x in the interval that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

28. Find all the values x in the interval that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π Q I Q III

29. Find all the values x in the interval that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

30. Find all the values x in the interval that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

31. Find all the values x in the interval that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π Q IVQ II

32. Find all the values x in the interval that satisfy the equation. -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

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. Sketch the graph of y = sin x + 1 -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

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. Sketch the graph of y = cos x - 2 -π 2 -π -3π 2 -2π π2π2 π 3π 2 2π

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