Evaluate: COS(ARCSIN(ROOT3/2))?

Question

what do u get for that question...what about this one
sin(arctan(4/7))

Answer 1

COS(ARCSIN(ROOT3/2)) = 1/2
sin(arctan(4/7)) = 4/√65
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Ideas: Use two right triangle relations:1 - √3 - 2, and 4 - 7 - √65

Answer 2

COS(ARCSIN(ROOT3/2))
The angle whos sine is (sqrt(3))/2 is 60 degrees
cos(60) = .5

sin(arctan(4/7)) = .496139
Draw a right triangle with short leg =4 and long leg = 7.
The hypotenuse = sqrt(65)
So sin x = 4/sqrt(65) = .496139

Answer 3

cos(arc sin sqrt(3/2)
let arc sin sqrt(3/2) =x
sin x = sqrt(3/2
squaring
sin^2x = 3/2
1- cos^2x = 3/2
cos^2x = 1- 3/2
= -1/2
cosx = sqrt(1/2) (since x is in first quadrant only positive value is taken
cos(arcsinsqrt(3/2) = cos x = sqrt(1/2)

2)
let arc tan(4/7) = y
tan y = 4/7
sin y/ cos y = 4/7
7 sin y = 4 cos y
squaring
49 sin^2 y = 16 cos^2 y
49 sin^2 y = 16(1-sin^2 y)
49 sin^2 y = 16 - 16 sin^2 y
65 sin^2 y = 16
sin^2y = 16/65
sin y = 4/sqrt(65)
sin(arctan4/7) sin y = 4/sqrt(65)