TUTORIAL (1) - Domain, Range, Zeros and Vertical Asymptotes of the 6 Basic Trigonometric Functions - Answers
- f(x) = sinx(x)
- domain : (-? , +?)
- range : [-1 , 1]
- period : 2?
- zeros at x = k?, k is an integer (k = 0 , ± 1 , ± 2 , ...).
- f(x) = cos(x)
- domain : (-? , +?)
- range : [-1 , 1]
- period : 2?
- zeros at x = ?/2+ k?, k is an integer (k = 0 , ± 1 , ± 2 , ...).
- f(x) = tan(x)
- domain : all real numbers except ?/2 + k? where k is an integer.
- range : (-? , +?)
- period : ?
- zeros at x = k?, k is an integer (k = 0 , ± 1 , ± 2 , ...). (Note: tan(x) and sin(x) have the same zeros because tan(x) = sin(x) / cos(x))
- vertical asymptotes at x = ?/2 + k?, k is an integer.
- f(x) = cot(x)
- domain : all real numbers except k? where k is an integer.
- range : (-? , +?)
- period : ?
- zeros at x = ?/2+ k?, k is an integer (k = 0 , ± 1 , ± 2 , ...). (Note: cot(x) and cos(x) have the same zeros because cot(x) = cos(x) / sin(x))
- vertical asymptotes at x = k?, k is an integer.
- f(x) = sec(x)
- domain : all real numbers except ?/2 + k? where k is an integer.
- range : (-? , -1] U [1 , +?)
- period : 2?
- sec(x) has no zeros.
- vertical asymptotes at x = ?/2 + k?, k is an integer.
- f(x) = csc(x)
- domain : all real numbers except k? where k is an integer.
- range : (-? , -1] U [1 , +?)
- period : 2?
- csc(x) has no zeros.
- vertical asymptotes at x = k?, k is an integer.
TUTORIAL (2) - Relationship Between Basic Trigonometric Functions
- sin(x) = cos(x-&pi/2) , cos(x) = sin(x+&pi/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.
- 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.
- sin(x) = cos(x-?/2) , sec(x) = 1/cos(x) , csc(x) = 1/cos(x-?/2) , tan(x) = cos(x-?/2) / cos(x) ,
cot(x) = cos(x) / cos(x-?/2)
- cos(x) = sin(x+?/2) , sec(x) = 1/sin(x+?/2) , csc(x) = 1/sin(x) , tan(x) = sin(x) / sin(x+?/2) ,
cot(x) = sin(x+?/2) / sin(x)
More references and links related to trigonometric functions and their properties.
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