- Evaluate (A) Arccos(1) (C) Arcsin(1/2) (B) Aresin(0) (D) Arctan(-1)
The Correct Answer and Explanation is :
Let’s evaluate each of the given inverse trigonometric expressions and provide a detailed explanation for each:
(A) Arccos(1):
he arccosine function, denoted as arccos(x) or cos⁻¹(x), returns the angle θ whose cosine is x, with θ in the range [0, π] radians.or arccos(1), we seek θ such that cos(θ) = 1.he cosine of 0 radians is 1, and within the specified range, this is the only solution.herefore, arccos(1) = 0 radians.
(B) Aresin(0):
t appears there might be a typographical error, and “Aresin” is likely intended to be “Arcsin.”he arcsine function, denoted as arcsin(x) or sin⁻¹(x), returns the angle θ whose sine is x, with θ in the range [-π/2, π/2] radians.or arcsin(0), we seek θ such that sin(θ) = 0.he sine of 0 radians is 0, and within the specified range, this is the only solution.herefore, arcsin(0) = 0 radians.
(C) Arcsin(1/2):
or arcsin(1/2), we seek θ such that sin(θ) = 1/2.ithin the range [-π/2, π/2], the angle that satisfies this condition is π/6 radians (or 30 degrees), since sin(π/6) = 1/2.herefore, arcsin(1/2) = π/6 radians.
(D) Arctan(-1):
he arctangent function, denoted as arctan(x) or tan⁻¹(x), returns the angle θ whose tangent is x, with θ in the range (-π/2, π/2) radians.or arctan(-1), we seek θ such that tan(θ) = -1.ithin the specified range, the angle that satisfies this condition is -π/4 radians (or -45 degrees), since tan(-π/4) = -1.herefore, arctan(-1) = -π/4 radians.
Explanation:
nverse trigonometric functions are essential in mathematics as they allow us to determine angles when given specific trigonometric ratios.ach inverse function corresponds to a primary trigonometric function:
- Arcsin(x): nverse of the sine function; returns the angle whose sine is x.- Arccos(x): nverse of the cosine function; returns the angle whose cosine is x.- Arctan(x): nverse of the tangent function; returns the angle whose tangent is x.
hese functions have specific domains and ranges to ensure they are well-defined and yield unique principal values: - Arcsin(x): omain: [-1, 1], Range: [-π/2, π/2]- Arccos(x): omain: [-1, 1], Range: [0, π]- Arctan(x): omain: All real numbers, Range: (-π/2, π/2)
nderstanding these domains and ranges is crucial for correctly evaluating inverse trigonometric functions.or instance, while multiple angles can have the same sine or cosine values due to the periodic nature of trigonometric functions, the inverse functions are restricted to specific ranges to provide a unique output.
n practical applications, inverse trigonometric functions are used in fields such as physics, engineering, and computer science to solve problems involving angles and distances.or example, they are employed in calculating angles in navigation, analyzing wave functions, and in algorithms that require angle computations.
y mastering the evaluation of these functions and understanding their properties, one can solve a wide array of problems involving angles and their corresponding trigonometric ratios.