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Question
If sinθ = `(-4)/5` and θ lies in the third quadrant then the value of `cos theta/2` is ______.
Options
`1/5`
`-1/sqrt(10)`
`-1/sqrt(5)`
`1/sqrt(10)`
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Solution
If sinθ = `(-4)/5` and θ lies in the third quadrant then the value of `cos theta/2` is `-1/sqrt(5)`.
Explanation:
Given that: sinθ = `-4/5`, θ lies in third quadrant
cosθ = `sqrt(1 - sin^2 theta)`
= `sqrt(1 - (- 4/5)^2`
= `sqrt(1 - 16/25)`
= `sqrt(9/25)`
= `(+3)/(-5)`
∴ cosθ = `- 3/5`, θ lies in the third quadrant.
cosθ = `2cos^2 theta/2 - 1` ......`[because pi < theta < (3pi)/2, therefore pi/2 < theta/2 < (3pi)/4]`
⇒ `(-3)/5 = 2cos^2 theta/2 - 1`
⇒ `2cos^2 theta/2 = 1 - 3/5 = 2/5`
⇒ `cos^2 theta/2 = 2/(5 xx 2) = 1/5`
⇒ `cos theta/2 = +- 1/sqrt(5)`
⇒ `cos theta/2 = - 1/sqrt(5)` .......`[because pi/2 < theta/2 < (3pi)/4]`
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