Let O be the origin, and OX→,OY→,OZ→ be three unit vectors in the directions of the sides QR→,RP→,PQ→ . respectively, of a triangle PQR . |OX→×OY→|= If the triangle PQR varies, then the minimum value of cos⁡(P+Q)+cos⁡(Q+R)+cos⁡(R+P)

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sin⁡(P+R)

sin⁡2R

sin⁡(P+Q)

sin⁡(Q+R)

-32

-53

answer is , .

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Detailed Solution

OX→=QR→QR→;OY→=RP→RP→OX→×OY→=QR→×RP→QR→RP→=sinQR^P=sinR=sinP+QcosP+Q+cosQ+R+cosR+P=−cosP+cosQ+cosR≥−32 as cos⁡P+cos⁡Q+cos⁡R≤32 ∀P+Q+R=π

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