Table: Domains and Ranges of Trigonometric and Inverse Functions
The concept of domain and range of trigonometric functions plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding the domain and range helps you solve trigonometric equations, sketch graphs, and avoid common errors in calculations, especially in competitive exams like JEE, NEET, and boards.
What Is Domain and Range of Trigonometric Functions?
A domain and range of trigonometric functions refers to the sets of input (x-values) and output (y-values) where sine, cosine, tangent, cotangent, secant, and cosecant functions are defined. You’ll find this concept applied in areas such as solving triangles, graphing periodic curves, and real-world measurement problems.
Summary Table: Domain & Range of Trigonometric Functions
| Function | Domain | Range | Period |
|---|---|---|---|
| Sine, sin(x) | All real numbers (−∞, ∞) | [−1, 1] | 2π |
| Cosine, cos(x) | All real numbers (−∞, ∞) | [−1, 1] | 2π |
| Tangent, tan(x) | All real numbers except x ≠ (2n+1)π/2 | (−∞, ∞) | π |
| Cosecant, csc(x) | x ≠ nπ | (−∞,−1] ∪ [1,∞) | 2π |
| Secant, sec(x) | x ≠ (2n+1)π/2 | (−∞,−1] ∪ [1,∞) | 2π |
| Cotangent, cot(x) | x ≠ nπ | (−∞, ∞) | π |
Key Formula for Domain and Range of Trigonometric Functions
For any trigonometric function f(x):
- Domain = All values of x for which f(x) is defined
- Range = All possible y = f(x) outputs
For example, for sine: \( y = \sin(x),\ \text{Domain: } x \in (−∞, ∞),\ \text{Range: } y \in [−1, 1] \).
Step-by-Step Illustration: How to Find Domain and Range (tan x Example)
- Start with the function: \( y = \tan(x) \)
- Remember: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
- Check where denominator ≠ 0: \( \cos(x) ≠ 0 \)
- \( \cos(x) = 0 \) at \( x = (2n+1)\frac{\pi}{2} \), where n is an integer
- So, domain is all real x except \( (2n+1)\frac{\pi}{2} \)
- Range: values tan(x) can take for allowed inputs = all real numbers (−∞, ∞)
Domain and Range of Inverse Trigonometric Functions
| Function | Domain | Range |
|---|---|---|
| \( \sin^{-1}x \) (arcsin x) | [−1, 1] | [−π/2, π/2] |
| \( \cos^{-1}x \) (arccos x) | [−1, 1] | [0, π] |
| \( \tan^{-1}x \) (arctan x) | (−∞, ∞) | (−π/2, π/2) |
| \( \sec^{-1}x \) (arcsec x) | (−∞,−1] ∪ [1,∞) | [0, π]\ (excluding π/2) |
| \( \csc^{-1}x \) (arccsc x) | (−∞,−1] ∪ [1,∞) | [−π/2, π/2]\ (excluding 0) |
| \( \cot^{-1}x \) (arccot x) | (−∞, ∞) | (0, π) |
Common Mistakes and Exam Tips
- Forgetting to exclude undefined x-values in tangent, cot, sec, and csc.
- Mixing up the range for inverse trigonometric functions—always check if your answer is within the allowed range.
- Assuming all trigonometric functions have the same range as sine and cosine (−1 to 1), which is incorrect for tan, cot, sec, and csc.
- Not using the principal value branch for inverse functions in competitive exams.
- Skipping the periodicity—especially when dealing with multiple cycles in graphs.
Classroom Tip
A handy way to remember ranges:
- sin and cos: Always between -1 and 1
- tan, cot: All real numbers
- sec, csc: Outside [-1, 1]
Try These Yourself
- Find the domain and range of \( y = 2\sin(x) + 1 \).
- What is the range of \( y = \sec(x) \)?
- Determine the domain of \( y = \cot(x) \).
- For which values of x is \( \tan(x) \) not defined?
Step-by-Step Example Solution
Find the domain and range of \( y = \sin(x) - 3 \):
1. Start with sin(x): Range is [−1, 1]2. Subtract 3 from each value: [−1-3, 1-3] ⇒ [−4, −2]
3. Domain is not affected by shifting: All real numbers
4. **Final Answer:** Domain: all real numbers, Range: [−4, −2]
Relation to Other Concepts
Mastery of domain and range of trigonometric functions helps you with Trigonometric Identities and is essential for solving trigonometric equations and graph sketching. It’s also the entry point for understanding inverse trigonometric functions in higher classes.
We explored domain and range of trigonometric functions—from definition and key tables to stepwise examples, mistakes to avoid, and quick tricks for remembering results. Continue practicing with Vedantu’s trigonometric functions resources to become confident in solving all trigonometry domain/range problems!
FAQs on Domain and Range of Trigonometric Functions
1. What is the domain and range of trigonometric functions?
The domain of a trigonometric function represents all possible input values (usually angles) for which the function is defined. The range represents all possible output values the function can produce. For example, the sine function (sin x) has a domain of all real numbers (-∞, ∞) because you can input any angle. Its range is [-1, 1] because the output of sin x always falls between -1 and 1, inclusive.
2. What are the domains and ranges of the six main trigonometric functions (sin, cos, tan, cot, sec, csc)?
Here's a summary table of the domains and ranges for the six main trigonometric functions:
| Function | Domain | Range |
|---|---|---|
| sin x | (-∞, ∞) | [-1, 1] |
| cos x | (-∞, ∞) | [-1, 1] |
| tan x | x ≠ (2n + 1)π/2, n ∈ Z | (-∞, ∞) |
| cot x | x ≠ nπ, n ∈ Z | (-∞, ∞) |
| sec x | x ≠ (2n + 1)π/2, n ∈ Z | (-∞, -1] ∪ [1, ∞) |
| csc x | x ≠ nπ, n ∈ Z | (-∞, -1] ∪ [1, ∞) |
Where Z represents the set of integers.
3. How do I find the domain of a trigonometric function?
To find the domain, identify values where the function is undefined. This usually involves:
- Fractions: Set the denominator not equal to zero and solve for x. These solutions are excluded from the domain.
- Square roots: The expression under the square root must be greater than or equal to zero. Solve the resulting inequality to find the permissible x values.
- Other functions: Consider the domains of any other functions within the trigonometric expression.
Always express the domain using interval notation or set notation.
4. How do I find the range of a trigonometric function?
Determining the range involves understanding the output values. Consider the following:
- Basic functions (sin, cos): Their ranges are [-1, 1].
- Reciprocal functions (sec, csc): Their ranges are (-∞, -1] ∪ [1, ∞).
- Tan and cot: Their ranges are (-∞, ∞).
- Transformations: Vertical shifts, stretches, and compressions will affect the range. Analyze how these transformations change the minimum and maximum output values.
Graphing the function can also help visually identify the range.
5. What are the domains and ranges of inverse trigonometric functions (arcsin, arccos, arctan, etc.)?
Inverse trigonometric functions have restricted domains and ranges to ensure they are one-to-one functions. Here's a table summarizing their principal values:
| Function | Domain | Range |
|---|---|---|
| arcsin x | [-1, 1] | [-π/2, π/2] |
| arccos x | [-1, 1] | [0, π] |
| arctan x | (-∞, ∞) | (-π/2, π/2) |
| arccot x | (-∞, ∞) | (0, π) |
| arcsec x | (-∞, -1] ∪ [1, ∞) | [0, π/2) ∪ (π/2, π] |
| arccsc x | (-∞, -1] ∪ [1, ∞) | [-π/2, 0) ∪ (0, π/2] |
6. Why are the domains of tan x and sec x restricted?
The tangent function (tan x = sin x / cos x) is undefined whenever cos x = 0, which occurs at odd multiples of π/2. Similarly, the secant function (sec x = 1 / cos x) is undefined at the same points because division by zero is undefined.
7. Why are the domains of cot x and csc x restricted?
The cotangent function (cot x = cos x / sin x) is undefined whenever sin x = 0, occurring at multiples of π. The cosecant function (csc x = 1 / sin x) is similarly undefined at these points due to division by zero.
8. How does periodicity affect the domain and range of trigonometric functions?
Periodicity means the function repeats its values after a specific interval. This affects the range only if the function is bounded (like sin and cos), implying that the range remains the same even though the function repeats. The domain, however, extends infinitely because the periodic nature ensures the function is defined for an infinite number of input values.
9. Can I use a graphing calculator to determine the domain and range?
Yes! Graphing calculators or online tools like Desmos can help visualize the function. Look at the x-values where the graph is defined (domain) and the y-values covered by the graph (range).
10. How do transformations (like shifts and stretches) affect the domain and range?
Vertical shifts change the range by shifting the minimum and maximum y-values. Horizontal shifts change the location where the function is defined but generally don't change the domain or range unless it causes asymptotes or affects the boundaries. Vertical stretches or compressions alter the range by scaling the output values. Horizontal stretches or compressions generally don't affect the range and can change the domain, but only to account for transformations of undefined points.
11. What are some common mistakes to avoid when working with the domain and range of trigonometric functions?
Common mistakes include:
- Forgetting to consider where the function is undefined (especially with fractions and square roots).
- Incorrectly determining the range of functions, particularly when dealing with reciprocal functions.
- Failing to account for transformations (shifts, stretches) when determining domain and range.
- Confusing domain and range values.
