Mathematics

Problem

Let

f(x)=\arcsin x

and

g(x)=\arccos x

for

x\in[-1,1]

. a. (i) On the same set of axes, sketch the graphs of

y=f(x)

and

y=g(x)

, indicating their endpoints. b. (ii) Solve algebraically the equation

\arcsin x=\arccos x

and give the exact coordinates of the intersection point of the two graphs. c. (iii) Using your sketches, state which of the two functions lies above the other on the intervals

(-1,\,1/\sqrt{2})

and

(1/\sqrt{2},\,1)

. d. (iv) Justify the result in part (iii) purely algebraically, without referring to the sketch.

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