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Question

If sinθ,cosθ and 3,2 lie on the same side of the line x+y=1, then θ lies between


A

0,π2

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B

0,π

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C

π4,π2

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D

0,π4

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Solution

The correct option is D

0,π4


Explanation for the correct option:

Step 1: Requirements for the solution

Since both, the given points lie on the same side of the line x+y=1. Let us call this line Q.

After applying the coordinates 3,2, we have

Q3,2=3+2-1=4>0

Then after applying the coordinates sinθ,cosθ in the line the result will be greater than zero because both points lie on the same side, then

Qsinθ,cosθ=sinθ+cosθ-1>0

Step 2: Finding where θ lies

As we know that sinθ+cosθ-1>0 then

sinθ+cosθ>1

Divide both the sides by 2,

sinθ12+12cosθ>12sinθcosπ4+sinπ4cosθ>12sinπ4=12andcosπ4=12

Now, apply the identity sin(A+B)=sin(A)cos(B)+sin(B)cos(A)

sinθ+π4>12

We know that this can only happen when sinA is greater than π4 but lesser than π2, where A=θ+π4.

π4<θ+π4<π2π4-π4<θ<π2-π40<θ<π4

Hence, the correct option is (D),


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