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प्रश्न
Find the general solutions of the following equation:
sin θ - cos θ = 1
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उत्तर
sin θ − cos θ = 1
∴ cos θ − sin θ = −1
∴ (1) cos θ − (1) sin θ = −1
`sqrt((1)^2 + (1)^2) = sqrt(1 + 1) = sqrt2`
dividing b. s. by `sqrt2`
∴ `1/sqrt2 costheta - 1/sqrt2 sintheta = -1/sqrt2`
∴ `cos pi/4 costheta - sin pi/4 sintheta = - cos pi/4`
∴ `cos"A" cos"B" - sin"A" sin"B" = cos"(A + B)"`
∴ `cos(theta + pi/4) = cos (pi - pi/4) ...(∵ - costheta = cos(pi - theta))`
∴ `cos (theta + pi/4) = cos (3pi)/4`
cos θ = cos α ⇒ θ = 2n π ± α, n ∈ 2
∴ `theta + pi/4 = 2npi +- (3pi)/4, n ∈ 2`
∴ `theta = 2npi +- (3pi)/4 - pi/4, n ∈ 2`
∴ `theta = 2n pi + (3pi)/4 - pi/4 or theta = 2npi - (3pi)/4 - pi/4, n ∈ 2`
∴ `theta = 2npi + (2pi)/4 or theta = 2npi - (4pi)/4 - pi/4, n ∈ 2`
∴ `theta = 2npi + pi/2 or theta = 2npi - pi n ∈ 2`
∴ These are required general solutions.
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