Question

Given a and b are first quadrant angles, sin a=5/13 and cos b=3/5 evaluate tan(a-b)1) 33/562) -33/563) -21/56

Answer 1

The angles given are

`\sin a=(5)/(13),\cos b=(3)/(5)`

Determine the value of cos a

`\cos a=\sqrt[]{1-(25)/(169)}=(12)/(13)`

Determine the value of sin b

`\sin b=\sqrt[]{1-(9)/(25)}=(4)/(5)`

The value of tan a is

`\tan \text{ a=}((5)/(13))/((12)/(13))=(5)/(13)*(13)/(12)=(5)/(12)`

The value of tan b is

`\tan b=((4)/(5))/((3)/(5))=(4)/(3)`

The value of tan(a-b) is

`\tan (a-b)=(\tan a-tanb)/(1+\tan a\tan b)`
`\tan (a-b)=((5)/(12)-(4)/(3))/(1+(5)/(12)*(4)/(3))=((5-16)/(12))/(1+(5)/(9))=((-11)/(12))/((14)/(9))=-(11)/(12)*(9)/(14)=-(11)/(4)*(3)/(14)=-(33)/(56)`
`\tan (a-b)=-(33)/(56)`

Hence the correct option is 2.