Solutions to Questions on Trigonometric Functions

Here are given the solutions and answers to the questions in Graphs of Basic Trigonometric Functions. For each trigonometric function is given the domain, range, period and asymptotes (if any). These properties are necessary to understand the graphs of the trigonometric functions. The relationship between trigonometric functions are also explored.

TUTORIAL (1) - Domain, Range, Zeros and Vertical Asymptotes of the 6 Basic Trigonometric Functions - Answers

  1. f(x) = sinx(x)
    • domain : (-∞ , +∞)
    • range : [-1 , 1]
    • period : 2π
    • zeros at x = kπ, k is an integer (k = 0 , ~+mn~ 1 , ~+mn~ 2 , ...).


  2. f(x) = cos(x)
    • domain : (-∞ , +∞)
    • range : [-1 , 1]
    • period : 2π
    • zeros at x = π/2+ kπ, k is an integer (k = 0 , ~+mn~ 1 , ~+mn~ 2 , ...).


  3. f(x) = tan(x)
    • domain : all real numbers except π/2 + kπ where k is an integer.
    • range : (-∞ , +∞)
    • period : π
    • zeros at x = kπ, k is an integer (k = 0 , ~+mn~ 1 , ~+mn~ 2 , ...). (Note: tan(x) and sin(x) have the same zeros because tan(x) = sin(x) / cos(x))
    • vertical asymptotes at x = π/2 + kπ, k is an integer.


  4. f(x) = cot(x)
    • domain : all real numbers except kπ where k is an integer.
    • range : (-∞ , +∞)
    • period : π
    • zeros at x = π/2+ kπ, k is an integer (k = 0 , ~+mn~ 1 , ~+mn~ 2 , ...). (Note: cot(x) and cos(x) have the same zeros because cot(x) = cos(x) / sin(x))
    • vertical asymptotes at x = kπ, k is an integer.


  5. f(x) = sec(x)
    • domain : all real numbers except π/2 + kπ where k is an integer.
    • range : (-∞ , -1] U [1 , +∞)
    • period : 2π
    • sec(x) has no zeros.
    • vertical asymptotes at x = π/2 + kπ, k is an integer.


  6. f(x) = csc(x)
    • domain : all real numbers except kπ where k is an integer.
    • range : (-∞ , -1] U [1 , +∞)
    • period : 2π
    • csc(x) has no zeros.
    • vertical asymptotes at x = kπ, k is an integer.

TUTORIAL (2) - Relationship Between Basic Trigonometric Functions

  1. sin(x) = cos(x-&pi/2) , cos(x) = sin(x+&pi/2)

  2. Since csc(x) = 1 / sin(x), the values that make sin(x) = 0 will create a division by zeros of the form 1/0 for csc(x) meaning that there is a vertical asymptotes at the same x values.

  3. Since sec(x) = 1 / cos(x), the values that make cos(x) = 0 will create a division by zeros of the form 1/0 for sec(x) meaning that there is a vertical asymptotes at the same x values.

  4. sin(x) = cos(x-π/2) , sec(x) = 1/cos(x) , csc(x) = 1/cos(x-π/2) , tan(x) = cos(x-π/2) / cos(x) ,
    cot(x) = cos(x) / cos(x-π/2)

  5. cos(x) = sin(x+π/2) , sec(x) = 1/sin(x+π/2) , csc(x) = 1/sin(x) , tan(x) = sin(x) / sin(x+π/2) ,
    cot(x) = sin(x+π/2) / sin(x)

More references and links related to trigonometric functions and their properties.