Trigonometric functions  Graphing and interpretation.
Test Yourself 1.
Given an equation, describe the features (include references to shifts and dilations). 
1. Describe the main features for the curve y = 3cos 2x. 
2. Describe the main features for the curve y = 2 + sin x.  
3. Describe the main features for the curve y = 2  3 cos 4x.  
4. Describe the main features for the curve .  
5. Describe the main features for the curve  
Given a graph, interpret the features (include references to shifts and dilations). 
6.
The graph above can be represented by an equation of the form Find the values of a and n. Answer.a = 2 and n = π/2. 
7. The equation of the following graph is written in the form y = A + B cos(Cx  D). Determine the values of A, B, C and D. Answer.A = 4, B = 2, C = 1 and D = 0. 

8. The equation of the following graph is written in the form y = A + B tan(Cx + D). Determine the values of A, B, C and D. Answer.A = 0, B = 3, C = 2 and D = 0. 

9. The equation of the following graph is written in the form y = A + B sin(Cx + D). Determine the values of A, B, C and D. Answer.A = 3, B = 2, C = 1 and D = π/3. 

10. The graph of D = A + Bcos Ct is given below.
Answer.A = 3, B = 2, C = π. 

11. What is the equation for the curve shown in the diagram below? Explain your reasons for your answer. 

Sketch graphs with transformations.  12. Sketch y = cos t + 2 for t:[0, 2π]. 
13. For the function y = 2 cos x


14. For the function y = 2 sin 3x;


15. Draw sketches, on the same axes, of the curves
and in the domain of x:[0, 4]. 

16. Sketch the curve y = 1  2 sin 2t for 0 ≤ t ≤ 2π.  
17. Sketch the curve y = 3 + 2 cos πt for 0 ≤ t ≤ 2π.  
18. Sketch the curve y = 2 + 3tan 2t for π ≤ t ≤ π.  
19. Sketch the curve y = 2  3 sin (t  ) for 0 ≤ t ≤ 2π.  
20. (i) Sketch the graph of y = 1  2cos x for 0 ≤x ≤ 2π. Clearly indicate the end points of the curve in its given domain as well as its turning points. (ii) Use your graph to solve 1  2cos x = 0 in the given domain. 

21. (i) Sketch the curves y = sin x and y = cos x on the same axes for 0 ≤ x ≤ 2π.
(ii) By adding ordinates, develop the graph of y = sin x + cos x. 

Graphs of reciprocal functions.  22. Sketch the curve y = 2 cosec 2x for 0° ≤ x ≤ 360°. 
23. Sketch the curve y = cot 2t for 0° ≤ t ≤ 360°.  
24. Sketch the curve y = sec x + 2 for 0 ≤x ≤ 2π.  
25. Sketch the curve y = sec^{2} x for 0 ≤x ≤ 2π.  
26. Sketch the curve y = 2 cot^{2} t for π ≤ t ≤ π.  
27. Sketch the curve y = 2  cosec x for 3π/2 ≤ x ≤ 3π/2.  
Given features, draw the graph.  28. Draw a sine graph which has a maximum value of 7 and a minimum value of 1 and has two patterns in its domain of 0 ≤x ≤ 2π. 
29. Draw a tan graph using the domain 0 ≤x ≤ 2π with the function having consecutive values of 0 at x = 0 and at x = π/4 and also a value of 5 at
x = π/8. 

30. Draw a sec graph which has minimum values of 2 at t = 0 and at t = 2π and a maximum value of 2 at t = π. Use the domain of 2π ≤ t ≤ 2π. 

31. Draw a cosine graph having a minimum value of 4 at x = 0, a maximum value of 0 and having two patterns in its domain of 0 ≤x ≤ 2π. 

Determining the number of solutions.  32. (i) Sketch the graph of y = 2cos 2x for π ≤ x ≤ π.

33. (i) Sketch the graphs of y = 3 cos 2θ and y = 1 for 0 ≤ θ ≤ π.
(ii) Use your graphs to obtain two approximate solutions to the equation. (iii) Solve the equation 3 cos 2θ = 1 using the normal technique for 0 ≤ θ ≤ π (answer to 3 decimal places) and compare your answers with those in part (ii). 

34. (i) Sketch y = tan πx for 0 ≤ x ≤ 2.


35. (i) Sketch the graph of y = sin 2x for 0 ≤ x ≤ π using relevant subintervals.


36. (i) Draw a neat sketch of the curve y = 3sin 2x for 0 ≤ x ≤ 2π.
