21 Questions
What is the range of the function f(x) = cos(tan⁻¹(x))?
(0, 1]
What is the property of cos⁻¹(x) used in the given inequalities?
Cos⁻¹(x) is a decreasing function
What is the graph of the function y = cos(x) like in the interval [-π/2, π/2]?
An increasing curve
What is the range of tan⁻¹(x) in the given solution?
(-π/2, π/2)
What is the purpose of multiplying both sides of the inequality by 3?
To eliminate the fraction
What is the domain of the function f(x) = cos(tan⁻¹(x))?
All real numbers
What is the range of the function f(x) = sin x?
[-1, 1]
What is the domain of the function f(x) = cos^{-1} x?
[−1, 1]
What is the value of sin^{-1} (1/2)?
π/6
What is the property of the function f(x) = tan^{-1} x?
tan^{-1} x = -tan^{-1} (-x)
What is the graph of the function f(x) = sin^{-1} x?
A continuous curve from -π/2 to π/2
What is the range of the function f(x) = tan^{-1} x?
(-π/2, π/2)
What is the value of cos^{-1} (√3/2)?
π/3
What is the property of the function f(x) = cos^{-1} x?
cos^{-1} x = -cos^{-1} (-x)
What is the domain of the function f(x) = tan x?
(-∞, ∞)
What is the graph of the function f(x) = cos x?
A continuous curve from -π to π
What is the range of the function $f(x) = an x$?
(-π/2, π/2)
What is the value of $ an^{-1}(-3)$?
$-rac{π}{3}$
What is the value of $ an^{-1}(1) + rac{π}{2}$?
$rac{3π}{2}$
What is the value of $ an^{-1}(1) + rac{π}{2} + rac{π}{6}$?
$rac{5π}{3}$
What is the value of $ an^{-1}(1) + rac{3π}{2} + rac{π}{6}$?
$rac{7π}{3}$
Study Notes
Inverse Trigonometric Functions
Domain, Range & Graph of ITF
-
tan:
- Restricted Domain: (-π, π/2) ∪ (π/2, π)
- Range: ℝ
-
cot:
- Restricted Domain: (0, π)
- Range: ℝ
-
sec:
- Restricted Domain: (0, π) - π/2 - π
- Range: (-∞, -1] ∪ [1, ∞)
-
cosec:
- Restricted Domain: (-π/2, 0) ∪ (0, π/2)
- Range: (-∞, -1] ∪ [1, ∞)
Examples
- cot⁻¹(0) = π/2
- cot⁻¹(1) = π/4
- cot⁻¹(-1) = 3π/4
Properties of Inverse Trigonometric Functions
- sin⁻¹(-x) = -sin⁻¹(x)
- cos⁻¹(-x) = π - cos⁻¹(x)
- tan⁻¹(-x) = -tan⁻¹(x)
- cot⁻¹(-x) = π - cot⁻¹(x)
- sec⁻¹(-x) = π - sec⁻¹(x)
- cosec⁻¹(-x) = -cosec⁻¹(x)
Graphs of Inverse Trigonometric Functions
-
y = sin⁻¹(x)
- Domain: [-1, 1]
- Range: [-π/2, π/2]
-
y = cos⁻¹(x)
- Domain: [-1, 1]
- Range: [0, π]
-
y = tan⁻¹(x)
- Domain: ℝ
- Range: (-π/2, π/2)
Additional Examples
-
Find the value of tan⁻¹(3) and tan⁻¹(-3)
- Solution: use the properties of inverse trigonometric functions to find the values.
Solve for θ when tangent inverse of -3 is equal to θ, where tangent of θ is equal to -3. Trigonometry problem involving inverse tangent function.
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