The range of the arctan function, $$\arctan(x)$$, is $$(-\frac{\pi}{2}, \frac{\pi}{2})$$. The angle $$\theta$$ in this interval for which $$\tan(\theta) = \sqrt{3}$$ is $$\frac{\pi}{3}$$.

We evaluate each expression: A) $$\arcsin(0.5) = \frac{\pi}{6}$$. B) $$\arccos(0.5) = \frac{\pi}{3}$$. C) $$\arctan(1) = \frac{\pi}{4}$$. D) $$\arcsin(-1) = -\frac{\pi}{2}$$. Comparing the values, $$\frac{\pi}{6} \approx 0.524$$, $$\frac{\pi}{3} \approx 1.047$$, $$\frac{\pi}{4} \approx 0.785$$, and $$-\frac{\pi}{2} \approx -1.571$$. The greatest value is $$\frac{\pi}{3}$$.

For a function to have an inverse, it must be one-to-one. The standard restriction for the domain of $$\cos(x)$$ to define $$\arccos(x)$$ is $$[0, \pi]$$. On this interval, the cosine function is one-to-one (it passes the horizontal line test) and its range is $$[-1, 1]$$, which is the complete range of the cosine function.

The fundamental inverse function property states that $$f(f^{-1}(x)) = x$$ for all $$x$$ in the domain of the inverse function. For $$\arcsin(x)$$, this means $$\sin(\arcsin(x)) = x$$ for all $$x$$ in $$[-1, 1]$$. Choice B is only true when $$x$$ is in the range of $$\arcsin$$, which is $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$. Choices C and D represent different relationships that are not the fundamental inverse property.

The domain of an inverse function is the range of the original function. The range of $$f(x) = \sin(x)$$ on the restricted domain $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$ is $$[-1, 1]$$. Therefore, the domain of $$f^{-1}(x) = \arcsin(x)$$ is $$[-1, 1]$$.

This question tests understanding of inverse trigonometric functions, focusing on arccos to find an angle from a cosine ratio. Inverse trigonometric functions find the angle whose trigonometric function gives a specific value. For example, arccos(x) gives the angle θ such that cos(θ) = x. In this question, the problem provides cos(θ) = 5/13 ≈ 0.385, requiring calculation of the angle using arccos within the constraint 0° < θ < 90°. Choice B is correct because θ = arccos(5/13) ≈ 67.4°, which accurately calculates the angle in the first quadrant. Choice A (22.6°) is incorrect because it represents the complementary angle (90° - 67.4°), a common error when students confuse which angle in a right triangle has the given cosine value. Encourage students to remember that smaller cosine values correspond to larger angles in the first quadrant. Practice using the 5-12-13 Pythagorean triple to verify calculations. Watch for: confusion between an angle and its complement when working with trigonometric ratios.

This question tests understanding of inverse trigonometric functions, focusing on arctan to find a direction angle from a tangent ratio. Inverse trigonometric functions find the angle whose trigonometric function gives a specific value. For example, arctan(x) gives the angle θ such that tan(θ) = x. In this question, the problem provides a navigation scenario where tan(θ) = 9/7 ≈ 1.286, requiring calculation of the angle from east using arctan. Choice B is correct because θ = arctan(9/7) ≈ 52.1°, which accurately calculates the angle between the displacement vector and the east direction. Choice A (37.9°) is incorrect because it represents the complementary angle (90° - 52.1°), a common error when students confuse the angle from east with the angle from north. Encourage students to draw displacement vectors and clearly label reference directions. Practice interpreting navigation problems where angles are measured from different cardinal directions. Watch for: confusion about which side is opposite vs adjacent to the desired angle.

This question tests understanding of inverse trigonometric functions, focusing on arctan to find a ramp angle from a tangent ratio. Inverse trigonometric functions find the angle whose trigonometric function gives a specific value. For example, arctan(x) gives the angle θ such that tan(θ) = x. In this question, the problem provides a ramp where tan(θ) = 1.5/8.0 = 0.1875, requiring calculation of the angle above horizontal using arctan. Choice A is correct because θ = arctan(0.1875) ≈ 10.6°, which accurately calculates the relatively small ramp angle. Choice B (79.4°) is incorrect because it represents the complementary angle (90° - 10.6°), a common error when students confuse the ramp angle with the angle between the ramp and vertical. Encourage students to verify their answers make physical sense - a gentle ramp should have a small angle. Practice estimating angles before calculating: since tan(θ) < 1, the angle must be less than 45°. Watch for: unrealistic angle values in practical contexts.

This question tests understanding of inverse trigonometric functions, focusing on arctan to find an angle from a tangent ratio. Inverse trigonometric functions find the angle whose trigonometric function gives a specific value. For example, arctan(x) gives the angle θ such that tan(θ) = x. In this question, the problem provides an engineering bracket where tan(θ) = 9/12 = 0.75, requiring calculation of the incline angle using arctan. Choice A is correct because θ = arctan(0.75) ≈ 36.9°, which accurately calculates the angle above horizontal. Choice B (53.1°) is incorrect because it represents the complementary angle (90° - 36.9°), a common error when students confuse which angle in the right triangle matches the given ratio. Encourage students to always identify the angle location before applying inverse functions. Practice simplifying fractions before calculating (9/12 = 3/4) and verify results make sense for the physical situation. Watch for: calculator mode errors (degrees vs radians) and angle identification mistakes.

This question tests understanding of inverse trigonometric functions, focusing on arcsin to find an angle from a sine ratio. Inverse trigonometric functions find the angle whose trigonometric function gives a specific value. For example, arcsin(x) gives the angle θ such that sin(θ) = x. In this question, the problem provides a ladder scenario where sin(θ) = 16/20 = 0.8, requiring calculation of the angle using arcsin. Choice A is correct because θ = arcsin(0.8) ≈ 53.13°, which accurately calculates the angle of elevation within the constraint 0° < θ < 90°. Choice B (36.87°) is incorrect because it represents the complementary angle (90° - 53.13°), a common error when students confuse the angle of elevation with the angle at the top of the triangle. Encourage students to draw and label the right triangle clearly, identifying which angle corresponds to the given trigonometric ratio. Practice using inverse functions on calculators in degree mode, and always verify the answer makes physical sense in the context.