In a right triangle PQR with <Q=90, <P=theta  and <R=empty set. If PR=x+2 and RQ=x , find,
(i)square root of x plus 1 end root cot empty set
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Asked by Jerlin George | 20th Sep, 2015, 10:05: PM

Expert Answer:

G i v e n space t h a t space P Q R space i s space a space r i g h t space t r i a n g l e space r i g h t space a n g l e d space a t space Q. angle P R Q equals empty set space a n d space angle Q P R equals theta B y space a p p l y i n g space P y t h a g o r a s space T h e o r e m comma space w e space h a v e comma P R squared equals P Q squared plus Q R squared rightwards double arrow open parentheses x plus 2 close parentheses squared equals P Q squared plus x squared rightwards double arrow x squared plus 4 x plus 4 minus x squared equals P Q squared rightwards double arrow P Q squared equals 4 open parentheses x plus 1 close parentheses rightwards double arrow P Q equals 2 square root of x plus 1 end root left parenthesis i right parenthesis F r o m space t h e space t r i a n g l e space P Q R comma space c o t ϕ equals fraction numerator A d j a c e n t space s i d e space o f space a n g l e space ϕ over denominator O p p o s i t e space s i d e space o f space a n g l e space ϕ end fraction space space space space space space space equals fraction numerator Q R over denominator P Q end fraction space space space space space space space equals fraction numerator x over denominator 2 square root of x plus 1 end root end fraction rightwards double arrow c o t ϕ cross times square root of x plus 1 end root equals x over 2 left parenthesis i i right parenthesis F r o m space t h e space t r i a n g l e space P Q R comma space tan theta equals fraction numerator O p p o s i t e space s i d e space o f space a n g l e space theta over denominator A d j a c e n t space s i d e space o f space a n g l e space theta end fraction space space space space space space space equals fraction numerator Q R over denominator P Q end fraction space space space space space space space equals fraction numerator x over denominator 2 square root of x plus 1 end root end fraction rightwards double arrow tan theta cross times square root of x plus 1 end root equals x over 2 rightwards double arrow t a n theta cross times x to the power of 4 cross times square root of x plus 1 end root equals x over 2 cross times x to the power of 4 rightwards double arrow t a n theta cross times square root of x cubed plus x squared end root equals x over 2 cross times x to the power of 4 rightwards double arrow t a n theta cross times square root of x cubed plus x squared end root equals x to the power of 5 over 2 left parenthesis i i i right parenthesis F r o m space t h e space t r i a n g l e space P Q R comma space cos theta equals fraction numerator A d j a c e n t space s i d e space o f space a n g l e space theta over denominator h y p o t e n u s end fraction space space space space space space space equals fraction numerator P Q over denominator P R end fraction space space space space space space space equals fraction numerator 2 square root of x plus 1 end root over denominator x plus 2 end fraction

Answered by Vimala Ramamurthy | 20th Sep, 2015, 10:57: PM

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