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If sin theta =(15)/(17) " and cos ...

If sin `theta =(15)/(17) " and cos " phi = (12)/(13)` where `theta " and " phi` both lie in the first quadrant find the values of
`(i) " sin " (theta + phi) " " (ii) " cos " (theta - phi) " " (iii) " tan " ( theta + phi)`

Text Solution

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Given ` " sin " theta = (15)/(17) " and cos " phi = (12)/(13)`
Since `theta ` lies in the first quadrant we have
sin `theta gt 0 " cos " theta gt 0 " and tan " theta gt 0`
Again since ` phi ` lies in the first quadrant we have
`" sin " theta gt 0 " cos "phi gt 0 " and tan " phi gt0`
Now `" cos"^(2) theta =(1 -"sin"^(2) theta) =(1- (225)/(289)) =(64)/(289)`
`rArr " cos " theta = + sqrt((64)/(289)) =(8)/(17)`
And `"sin"^(2) phi =(1 -"cos"^(2) phi) =(1-(144)/(169)) =(25)/(169)`
`rArr " sin " phi = + sqrt((25)/(169))= (5)/(13)`
`:. " tan " theta = (" sin " theta)/("cos " theta)=((15)/(17) xx (17)/(8)) =(15)/(8) ` and
`" tan " phi = ("sin " phi)/(" cos " phi) =((5)/(13) xx (13)/(12)) = (5)/(12)`
Now `(i) " sin " (theta + phi) = " sin " theta " cos " phi = " cos " theta " sin " phi`
`=((15)/(17) xx (12)/(13)) + ((8)/(17) xx (5)/(13)) = ((180)/( 221) + (40)/( 221)) = (220)/( 221)`
` (ii) " cos " (theta - phi) = " cos " theta " cos " phi + " sin " theta " sin " phi`
` =((8)/(17) xx (12)/(13)) + ((15)/(17) xx (5)/(13)) = ((96)/(221) + (75)/(221)) = (171)/(221)`
` (iii) " tan " (theta + phi) = ("tan " theta + " tan " phi)/(1- " tan " theta " tan " phi)`
` =(((15)/(8) + (15)/(12)))/({-((15)/(8) xx (5)/(12)) } }= (((55)/(24)))/((1-(25)/(32))) =((55)/(24) xx (32)/(7))= (220)/(21)`
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