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Question
If tan x = `3/4` and `pi < x < (3pi)/2`, then find the value of sin `x/2` and cos `x/2`.
Solution
Given tan x = `3/4` and `pi < x < (3pi)/2`
Since x lies in the III quadrant, only tan and its reciprocal are positive.
sin x = `(-3)/5`, cos x = `(-4)/5`.
Now, sin `x/2 = sqrt((1 - cos x)/2) = sqrt(((1 - (-4/5))/2)`
`= sqrt((1 + 4/5)/2)`
`= sqrt(9/10)`
`= 3/sqrt10`
`cos x/2 = sqrt((1 + cos x)/2) = sqrt((1 + 4/5)/2)`
`= sqrt((1 - 4/5)/2) = sqrt(1/10) = 1/sqrt10`
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