The domain of a trigonometric function is the set of all input values (angles) for which the function is defined, while the range is the set of all possible output values of the function. In simple terms:
- Domain: Values of x that can be substituted into the function.
- Range: Values that the function can produce as output.
The domains and ranges of the six trigonometric functions are:
- Sine Function {Sin(θ)}: The sine function sin(θ) has a domain of all real numbers (−∞, ∞) and a range between -1 and 1 inclusive: -1 ≤ sin(θ) ≤ 1.
- Cosine Function {Cos (θ)}: The cosine function cos(θ) also has a domain of all real numbers and a range between -1 and 1: -1 ≤ cos(θ) ≤ 1.
- Tangent Function {Tan (θ)}: The tangent function, tan(θ) has a domain of all real numbers except the values where the cosine is 0. So, its domain is (-∞, ∞) excluding the values where cos(θ) = 0 which are odd multiples of π/2. Its range is all real numbers covering the entire real number line.
- Cotangent Function {Cot (θ)}: The cotangent function cot(θ) shares its domain with the tangent function excluding the points where tan(θ) = 0 (i.e., multiples of π). Its range like tan(θ) covers the entire real number line.
- Secant Function {Sec (θ)}: The secant function, sec(θ) has a domain of all real numbers except the values where cos(θ) = 0. Its range is also (-∞, ∞) excluding values where cos(θ) = 0.
- Cosecant Function {Cosec (θ)}: The cosecant function, cosec(θ) shares its domain with the sine function, excluding the values where sin(θ) = 0 (i.e., multiples of π). Its range is (-∞, ∞) excluding values where sin(θ) = 0.
Solved Examples
Example 1: Find the domain and range of y = sin(x) for all real numbers.
The domain and range of y will be as follows:
- Domain: R
- Range: [−1, 1]
Example 2: Determine the domain and range of y = 2cos(3x) for all real values of x.
The domain and range of y will be as follows:
- Domain: R
- Range: [−2, 2]
Example 3: Determine the value of y = cot(2x) for x = 8π
The value of y at x= 8π
y(8π) = cot(2× 8π) = 1
Example 4: Find the domain and range of y = −2tan(x) + 1 for all real values of x.
The domain and range of y will be as follows:
- Domain: R
- Range: (−∞, 1)
Example 5: Determine the value of y = cos( 2x) for x= 2π
The value of y at x= 2π
y(2π) = cos(4π) = 2
Practice Problems
Problem 1: Find the domain and range of the function: y = 3sin(x)
Problem 2: Determine the domain and range of: y = sec(x)
Problem 3: Find the value of the function: y = tan(3x) when x = π/12
Problem 4: Determine the domain and range of: y = 4cos(2x)−1 for all real values of x.
Problem 5: Find the value of: y=cosec(2x) when x = π/4x