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10. 方程

\arcsin x+\arccos y=n \pi \quad(n \in \mathbf{Z})

所表示的图解是 ( ). (A) (B...

Feb 26, 2024

10. 方程

\arcsin x+\arccos y=n \pi \quad(n \in \mathbf{Z})

所表示的图解是 ( ). (A) (B) (C) (D)

Solution by Steps

step 2

The sum of the two functions

\arcsin x + \arccos y

will be within the range

[-\frac{\pi}{2}, \frac{3\pi}{2}]

step 3

Since

n \in \mathbf{Z}

, and the equation is

\arcsin x + \arccos y = n\pi

, the only possible values for

n

that fall within the range are

n = 0

n = 1

step 4

For

n = 0

, the equation becomes

\arcsin x + \arccos y = 0

. This is only possible if

x = 0

and

y = 1

, which is a single point on the graph

step 5

For

n = 1

, the equation becomes

\arcsin x + \arccos y = \pi

. This is true for all

x

and

y

such that

x^2 + y^2 = 1

, which is the equation of a circle with radius 1

Key Concept

Sum of

\arcsin

and

\arccos

Explanation

The sum

\arcsin x + \arccos y = n\pi

represents specific geometric shapes depending on the value of

n

. For

n = 0

, it's a point, and for

n = 1

, it's a circle with radius 1.