The Algebra of Trigonometric Functions: Practice Exercises

Here are various Exercises to accompany the section The Algebra of Trigonometric Functions.  

Basic Trigonometric Identities

  1. Use the geometry of the unit circle:

     

    to match each of  the following function evaluations on the left with the corresponding exact value on the right.  Note: The same values may be used more than once.

    1. cos(p)
    2. sin(p)
    3. cos(-p/2)
    4. sin(-p/2)
    5. cos(2p/3) 
    6. sin(2p/3)
    7. cos(-p/4) 
    8. sin(-p/4)
    9. cos(5p/6)
    10. sin(5p/6)
    11. sin(7p/3)
    12. cos(7p/3)
    13. sin(-4p/5)
    14. cos(-4p/5)
    1. 1
    2. » 0.9
    3. » 0.8
    4. » 0.7
    5. » 0.6
    6. 1/2 = 0.5
    7. » 0.3
    8. 0
    9. - » -0.3
    10. -1/2 = -0.5
    11. -  » -0.6
    12. » -0.7
    13. - » -0.8
    14. - » -0.9
    15. -1

    Hint: First find the correct point on the unit circle, estimate the correct value to within 0.1, then use the approximate values to match up with the exact value.  Compare each answer with the values the table of exact values from the text 

    Solution.

  2. Use your answers to the previous Exercise and the definitions of the trigonometric functions to give exact values for the following:
    1. tan(p)
    2. sec(p)
    3. sec(-p/2)
    4. csc(-p/2)
    5. sec(2p/3) 
    6. cot(2p/3)
    7. csc(-p/4) 
    8. tan(-p/4)
    9. sec(5p/6)
    10. csc(5p/6)
    11. cot(7p/3)
    12. csc(7p/3)
    13. sec(-4p/5)
    14. tan(-4p/5)

    Solution.

  3. Use the indicated identity to solve for the indicated value:
    1. Use the Pythagorean identity to solve for sin(2p/5), starting from cos(2p/5) = .
    2. Use the double angle formula for sin to solve for sin(2p/5), starting from cos(p/5) = and sin(p/5) = .  Compare your answer to that from part a).
    3. Use the difference identity for cos to solve for cos(p/12) = cos(p/4 - p/6).
    4. Use the Pythagorean identity to solve for cos(p/12), starting from the value of sin(p/12) = from the text.  Compare your answer to that from part c).

    Solution.

  4. Follow the given instructions to derive even more accurate approximation formulas for sin and cos.
    1. Use the double angle formula for sin and the approximation formulas from the text for sin and cos (i.e., sin(t) » t and cos(t) » 1 - t2/2) to determine an approximation formula for sin(2t).
    2. Replace t by t/2 in your approximation formula from part a) to give a more elaborate approximation formula for sin(t).  By comparing the graphs of your approximation formula, t, and sin(t):

      explain whether or not your formula is a more accurate approximation than sin(t) » t.

    3. Use the double angle formula for cos and the approximation formula from the text for cos (i.e., cos(t) » 1 - t2/2) to determine another approximation formula for cos(2t).
    4. Replace t by t/2 in your approximation formula from part c) to give a more elaborate approximation formula for cos(t).  By comparing the graphs of your approximation formula, 1 - t2/2, and cos(t):

      explain whether or not your formula is a more accurate approximation than cos(t) » 1 - t2/2.

    Solution.

More Advanced Identities

  1. Use the indicated identity to solve for the indicated value:
    1. Use the half angle formula for cos to solve for cos(p/12), starting from cos(p/6) = .  Compare your answer with the value from the previous Exercise. 
    2. Use the half angle formula for tan solve for tan(p/12), from cos(p/6) = and sin(p/6) = 0.5.  Compare your answer with the value of tan(p/12) = sin(p/12)/cos(p/12), using the value of sin(p/12) = from the text and cos(p/12) from the previous Exercise.
    3. Use the half angle formula for cos to solve for cos(p/5), starting from cos(2p/5) = .  Compare your answer with the value given in the table from the text. 
    4. Use the half angle formula for tan to solve for tan(p/5), from cos(2p/5) = and sin(2p/5) = from a previous Exercise.  Compare your answer with the value of tan(p/5) = sin(p/5)/cos(p/5) = .

    Solution.

  2. Follow the given instructions to derive more trigonometric identities for tan, cot, sec, and csc.
    1. Divide both sides of the Pythagorean identity by cos2 and simplify to show that sec2(t) = 1 + tan2(t).
    2. Starting with the equation tan(t - s) = sin(t - s)/cos(t - s), use the difference formulas for sin and cos to give a difference formula for tan in terms of sin and cos.
    3. Divide the numerator and denominator of your formula from part b) by cos(t)cos(s), and simplify to give a difference formula for tan(t - s) in terms of tan(t) and tan(s).
    4. Divide the numerator and denominator of your formula from part b) by sin(t)sin(s), and simplify to give a difference formula for tan(t - s) in terms of cot(t) and cot(s).
    5. Use your formula from part d) to verify the cofunction identity tan(p/2 - t) = cot(t).  
    6. Starting from the equation , multiply numerator and denominator by 2cos(t/2), and apply the double angle identities for sin and cos to derive the second half angle formula for tan:

    Solution.

  3. Follow the given instructions to derive a triple angle formula for cos.
    1. Use the sum formula for cos to express cos(3t) = cos(2t + t) in terms of sin and cos of 2t and t.
    2. Starting with your identity from part a), use the double angle formulas for sin and cos to write this entirely in terms of sin(t) and cos(t).
    3. Use the Pythagorean identity to eliminate the factor of sin2(t) to give a formula for cos(3t) in terms of cos(t) alone.

    Solution.

Go to Graphing and Trigonometric Functions .

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