Find the value of sin (a+b) if tan a=4/3 and cot b= 5/12, 0^degrees<a<90^degrees and 0^degrees<b<90^degrees ?

2 Answers
May 1, 2018

sin(a+b)=56/65

Explanation:

Given, tana=4/3 and cotb=5/12

rarrcota=3/4

rarrsina=1/csca=1/sqrt(1+cot^2a)=1/sqrt(1+(3/4)^2)=4/5

rarrcosa=sqrt(1-sin^2a)=sqrt(1-(4/5)^2)=3/5

rarrcotb=5/12

rarrsinb=1/cscb=1/sqrt(1+cot^2b)=1/sqrt(1+(5/12)^2)=12/13

rarrcosb=sqrt(1-sin^2b)=sqrt(1-(12/13)^2)=5/13

Now, sin(a+b)=sina*cosb+cosa*sinb

=(4/5)(5/13)+(3/5)*(12/13)=56/65

May 1, 2018

sin(a+b)=56/65

Explanation:

Here,

0^circ < color(violet)( a ) < 90^circ=>I^(st)Quadrant=>color(blue)(All, fns. > 0.

0^circ < color(violet)(b) < 90^circ=>I^(st)Quadrant=>color(blue)(All,fns. > 0

So,

0^circ < color(violet)(a+b) < 180^circ=>I^(st) and II^(nd) Quadrant

=>color(blue)(sin(a+b) > 0

Now,

tana=4/3=>seca=+sqrt(1+tan^2a)=sqrt(1+16/9)=5/3

:.color(red)(cosa)=1/seca=color(red)(3/5

=>color(red)(sina)=+sqrt(1-cos^2a)=sqrt(1-9/25)=color(red)(4/5

Also,

cotb=5/12=>cscb=+sqrt(1+cot^2b)=sqrt(1+25/144)=13/12

:.color(red)(sinb)=1/cscb=color(red)(12/13

=>color(red)(cosb)=+sqrt(1-sin^2b)=sqrt(1-144/169)=color(red)(5/13

Hence,

sin(a+b)=sinacosb+cosasinb

=>sin(a+b)=4/5xx5/13+3/5xx12/13

sin(a+b)=20/65+36/65=56/65