Exponential and Trigonometric Functions: Practice Exercises

Here are various Exercises to accompany the section Exponential and Trigonometric Functions.  

Complex Numbers, Exponentials, and the Unit Circle

1. Plot the points which are t units around the unit circle:

    

   for the following values of t.  Remember: You may either think of t as measuring the distance around the circle, or the radian measure of the angle with the positive horizontal axis.

   1. t = p
   2. t = p/2
   3. t = p/3
   4. t = p/4
   5. t = p/5
   6. t = p/6
   7. t = 7p/6
   8. t = -3p/4
   9. t = -9p/2
   10. t = 2p/3

   Hint: You may want to use our table of values for the circle function, C, as a guide.

   Solution.

2. Give your best visual estimate for the following values of C, to within 0.2 units in the real and complex parts:
   1. C(p)
   2. C(p/2)
   3. C(p/3)
   4. C(p/4)
   5. C(p/5)
   6. C(p/6)
   7. C(7p/6)
   8. C(-3p/4)
   9. C(-9p/2)
   10. C(2p/3)

   Hint: These are simply the values as complex numbers of the points you plotted in the previous Exercise.

   Solution.

Introduction to Trigonometric Functions

1. Using your plot from the previous Exercise your best visual estimate for the following values of cos or sin, to within 0.2 units:
   1. cos(p)
   2. sin(p/2)
   3. cos(p/3) 
   4. sin(p/3)
   5. cos(p/4) 
   6. sin(p/4)
   7. cos(p/6)
   8. sin(p/6)
   9. sin(7p/6)
   10. sin(-3p/4)
   11. cos(-9p/2)
   12. cos(2p/3)

   Hint: Remember that cos and sin are simply the real and complex parts of C.

   Solution.

2. Use your calculator, or the "x =" box in XFunctions, to verify your answers to the previous Exercise by obtaining more exact values for cos and sin.  

   Solution.

3. Use either your approximate or more exact answers to the previous Exercises and the defining formulas from the text to estimate the values for the following trigonometric functions:
   1. sec(p)
   2. csc(p/2)
   3. tan(p/3) 
   4. cot(p/3)
   5. tan(p/4) 
   6. csc(p/4)
   7. cot(p/6)
   8. csc(7p/6)
   9. tan(-3p/4)
   10. sec(-9p/2)
   11. sec(2p/3)

   Solution.

Go to The Algebra of Trigonometric Functions .

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Send questions or comments to jwicks@northpark.edu

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