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If " sin " theta =(3)/(5) " and cos ...

`If " sin " theta =(3)/(5) " and cos " phi =(-12)/(13) " where " theta " and " phi` both lie in the second 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 = (3)/(5) " and cos " phi = (-12)/(13)`
Since `theta ` lies in the second quadrant we have
Again since `phi ` lies in the second quadrant we have
`" sin " theta gt 0 " cos " theta lt 0 " and tan " theta lt0`
Again since `phi` lies in the second quadrant we have
`"sin "phi gt 0 " cos " phi lt 0 " and tan " phi lt0`
Now `" cos " theta =- sqrt(1-sin^(2) theta)=- sqrt(1-(9)/(25))=- sqrt((16)/(25) )=(-4)/(5)`
and `"tan " theta = ("sin " theta)/(" cos " theta) =(3)/(5) xx (-(5)/(4)) =(-3)/(4)`
Also `" sin " phi = + sqrt(1- " cos ^(2) phi)= + sqrt(1-(144)/(169)) =+ sqrt((25)/(169)) =+ (5)/(13)`
and tan `phi =("sin " phi)/("cos " phi) =(5)/(13) xx ((-13)/(12)) =(-5)/(12)`
`:. (i) " sin " (theta - phi) = "sin " theta " cos " phi - " cos " theta " sin " phi`
`={(3)/(5) xx((-12)/(13))}- {((-4)/(5)) xx (5)/(13)}= ((-36)/(65) + (20)/(65)) =(-16)/(65)`
`(ii) " cos " (theta + phi) = " cos " theta " cos " phi - "sin " theta " sin " phi`
`={((-4)/(5)) xx ((-12)/(13))}-{(3)/(5) xx (5)/(13)}= ((48)/(65) -(3)/(13)) =(33)/(65)`
`(iii) " tan " (theta - phi) =("tan " theta - "tan " phi)/(1+ " tan " theta " tan " phi)`
`=(((-3)/(4))-((-5)/(12)))/(1+{((-3)/(4)) xx ((-5)/(12))} }=(((-3)/(4)+(5)/(12)))/((1+(5)/(16))) =((-1)/(3)) xx (16)/(21) =(-16)/(63)`
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