mohitmahajan714

mohitmahajan714

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Sri Satya Sai University of Technology and Medical Sciences - SSSUTMS

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Finding p and q.

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MANI addagarla

Find p and q,ifx=āˆša(sinu+cosv), y=āˆša(cosuāˆ’sinv), z=1+sin(uāˆ’v)

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To find p and q, we need to express x, y, and z in terms of p and q.

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We have:

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x = āˆša(sinu + cosv) = āˆša(sin(p + q) + cos(p - q))

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= āˆša(sin p cos q + cos p sin q + cos p cos q - sin p sin q)

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= āˆša[(sin p cos q + cos p sin q) + (cos p cos q - sin p sin q)]

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= āˆša(sin(p+q) + cos(p+q))

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Therefore, we have:

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sin(p+q) = sinu + cosv

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cos(p+q) = cosu - sinv

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Squaring both equations and adding them, we get:

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sin^2(p+q) + cos^2(p+q) = (sinu + cosv)^2 + (cosu - sinv)^2

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1 = 2 + 2sinucosv - 2sinvcosu

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sinucosv + cosusinv = 1/2

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Multiplying the first equation by cos(p+q) and the second equation by sin(p+q), we get:

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x cos(p+q) = āˆša(sinu + cosv)(cosu - sinv) = āˆša(sinv cosu + sinu cosv) = yp

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y sin(p+q) = āˆša(cosu - sinv)(sinu + cosv) = āˆša(cosu sinv - sinu cosv) = -xq

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Adding these two equations, we get:

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xp - yq = āˆša(sinv cosu + sinu cosv)cos(p+q) - āˆša(cosu sinv - sinu cosv)sin(p+q)

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= āˆša[sin(p+q)cosu + sin(p+q)cosv + cos(p+q)sinu - cos(p+q)sinv]

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= āˆša[sin(u+v)cos(p+q) + cos(u-v)sin(p+q)]

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= āˆša[sin(p+q)(cosu cosv + sinu sinv) + sin(p+q)(sinu cosv - cosu sinv)]

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= āˆša[sin(p+q)cos(u-v) + cos(p+q)sin(u+v)]

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= āˆša[sin(p+q)z + cos(p+q)y]

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Thus, we have:

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xp - yq = āˆša[sin(p+q)z + cos(p+q)y]

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Solving for p and q may be difficult without additional information or constraints.

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