# Q. The number of ordered 4-tuple (x,y,z,w) (x,y,z,w `in` [0,10]) which satisfied the inequality, `(2)^(sin^2x) (3)^(cos^2y) (4)^(sin^2z) (5)^(cos^2w)`` ` `>= 120` A) 0 B) 144 C) 81 D) 64

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Expert Answers

sciencesolve | Certified Educator

You should notice that the product `2*3*4*5 = 120` , hence, using the definitions of sine and cosine functions, yields:

`{(sin x <= 1,sin^2 x <= 1),(cos y <= 1,cos^2 y <= 1 ),(sin z <= 1 ,sin^2 z <= 1), (cos w <= 1,cos^2 w <= 1):}`

Considering `sin x = 1 => x in [0,10] => x = {pi/2,2pi + pi/2}`

Considering `cos y = 1 => y in [0,10] => y= {0,2pi}`

Considering `sin z = 1 => x in [0,10] => z = {pi/2,2pi + pi/2}`

Considering `cos w = 1 => w in [0,10] => y= {0,2pi}`

Each angle x,y,z,w has two possible values, hence, the total number of possible values is of `N = 8*8 = 64`

**Hence, evaluating the total number of ordered 4-tuples (x,y,z,w) for the given inequality to hold, yields `N = 64` , hence, you need to select the answer `D) 64` .**