Exponential and Trigonometric Functions: Solutions

Here are some solutions to the 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
    Since a full rotation is 2p, we know that p is only halfway around the circle at -1.  Since p/2 is half of that, it is one-quarter of the way around (counter-clockwise) at i.  We can find t = p/3, t = p/4, t = p/5, and t = p/6, by dividing both semi-circles into thirds, quarters, fifths, and sixths, respectively:

     By continuing to divide the entire circle into the same size pieces, and by counting, we can find the other points.  For example, to find t = -3p/4, we would divide the entire circle into eight equal pieces (i.e., dividing each half-circle in fourths) and count clockwise (i.e., the negative direction) 3 steps.  Likewise, to find t = -9p/2, we divide the circle into quarters (can you see why?) and count 9 steps clockwise, until we reach the point -i (for the third time).

    Back to Exercises.

  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
    Dividing the intervals [-1, 0], [0, 1], [-i, 0], and [0, i] on the axes into fifths (since 0.2 = 1/5), we can estimate the values for each point:

     

    C(p) = -1, C(p/2) = i, C(p/3) » 0.5 + 0.9i, C(p/4) » 0.7 + 0.7i, C(p/5) » 0.8 + 0.6i, C(p/6) » 0.9 + 0.5i, C(7p/6) » -0.9 - 0.5i, C(-3p/4) » -0.7 - 0.7i, C(-9p/2) » -i, C(2p/3) » -0.5 + 0.9i.

    Back to Exercises.

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)

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

    Solution
    cos(p) = -1, sin(p/2) = 1, cos(p/3) » 0.5, sin(p/3) » 0.9, cos(p/4) » 0.7, sin(p/4) » 0.7, cos(p/6) » 0.9, sin(p/6) » 0.5, sin(7p/6) » -0.5, sin(-3p/4) » -0.7, cos(-9p/2) = 0, cos(2p/3) » -0.5.

    Back to Exercises.

  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
    Many of our approximations were, in fact, exactly correct!  For example, in the next section, we will be able to prove that cos(p/3) = 1/2.  For the others, we give more exact answers.  Overall, we have cos(p) = -1, sin(p/2) = 1, cos(p/3) = 0.5, sin(p/3) » 0.866025, cos(p/4) » 0.707107, sin(p/4) » 0.707107, cos(p/6) » 0.9, sin(p/6) = 0.5, sin(7p/6) = -0.5, sin(-3p/4) » -0.707107, cos(-9p/2) = 0, cos(2p/3) = -0.5 .  Notice how the values 0.866025 and 0.707107 keep appearing.  We will be able to compute these values exactly, as well.

    Back to Exercises.

  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
    sec(p) = 1/cos(p) = 1/(-1) = -1.
    csc(p/2) = 1/sin(p/2) = 1/1 = 1.
    tan(p/3) = sin(p/3)/cos(p/3) » 0.9/0.5 = 1.8, or more exactly, 0.866025/0.5 = 1.73205.
    cot(p/3) = cos(p/3)/sin(p/3) » 0.5/0.9 = 0.555..., or more exactly, 0.5/0.866025  » 0.57735.
    csc(p/4) = 1/sin(p/4) » 1/0.7 = 0.142857..., or more exactly 1/0.707107 » 1.41421. cot(p/6) = cos(p/6)/sin(p/6) » 0.9/0.5 = 1.8, or more exactly, 0.866025/0.5 = 1.73205. csc(7p/6) = 1/sin(7p/6) = 1/(-0.5) = -2.
    tan(-3p/4) = sin(-3p/4)/cos(-3p/4) = 1.  
    sec(-9p/2) = 1/cos(-9p/2) = 1/0 = undefined.
    sec(2p/3) = 1/cos(2p/3) = 1/(-0.5) = -2.
    Note: If you recognize the values of 1.41421... and 1.73205... as  and , respectively, you can deduce exact formulas for all the values in this and the previous Exercise.

    Back to Exercises.

Go to The Algebra of Trigonometric Functions.

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