maths solve

If sinθ + 2cosθ = 1, find the value of 2sinθ - cosθ.

-1

I don't know how to type the 'theta' symbol with my computer keyboard so I'll just use 'a' to denote the angle.

Given, sin a + 2 cos a = 1

Squaring both sides, we get (using (a+b)^2=a^2+b^2+2ab),

(sin a + 2 cos a)^2 = (1)^2

sin^2 a + 4 cos^2 a + 2*sin a*2 cos a = 1

Through an identity, sin^2 a+ cos^2 a=1 so sin^2 a= 1-cos^2 a and cos^2 a= 1-sin^2 a

Replacing (sin^2 a)  by (1-cos^2 a) and (cos^2 a) by (1-sin^2 a) in the original equation, we get

(1-cos^2 a) + 4(1-sin^2 a) + 4 sin a*cos a=1

1-cos^2 a + 4 - 4 sin^2 a + 4*sin a*cos a=1

Adding (-1) to both sides we get

4-4 sin^2 a- cos^2 a+4*sin a*cos a=0

-4 sin^2 a - cos^2 a+ 4* sin a* cos a= (-4)

Multipyling both sides by (-1),

4 sin^2 a + cos ^2 a- 4*sin a* cos a= 4

(2 sin a)^2 + (cos a)^2 -2*2 sin a* cos a= (2)^2

(2 sin a- cos a)^2 = (2)^2

2 sin a - cos a= 2,  which is the required answer.

2

i mean -1

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