Question

11. If cosA= 3 5 and sinB= 5 13 then the value of tan(A+B)value=​

Answer 1

Answer:

tan(A+B)=$\frac{252}{80}$

Step-by-step explanation:

Given,

        $CosA=\frac{3}{5} \\ \\ SinB=\frac{5}{13}$

As we know,

                  $Sin^{2} A+Cos^2A=1\\ \\ Sin^{2} A=1-Cos^2A}\\ \\ SinA=\sqrt{1-Cos^2A}} \\ \\ SinA=\sqrt{1-(\frac{3}{5})^2 } \\ \\ SinA=\sqrt{\frac{25-9}{25} } \\ \\ SinA=\frac{4}{5}$

Again,

                   $CosB=\sqrt{1-(Sin^2B)} \\ \\ =\sqrt{\frac{169-25}{169}} \\ \\ =\sqrt{\frac{144}{169} } \\ \\ =\frac{12}{13}$

               $tanA=\frac{\frac{4}{5} }{\frac{3}{5} } \\ \\ tanA=\frac{4}{3} \\ \\ \\ tanB=\frac{\frac{5}{13} }{\frac{12}{13} } \\ \\ =\frac{5}{12}$

tan(A+B)=$\frac{tanA+tanB}{1-tanA \cdot tanb} \\ \\ =\frac{\frac{4}{3}+\frac{5}{12} }{1-\frac{4}{3} \cdot\frac{5}{12} } \\ \\ =\frac{\frac{21}{12} }{\frac{20}{36} } \\ \\ =\frac{21 \cdot36}{12 \cdot 20} =\frac{252}{80}$

Answer 2

Given: cos A=3/5 and sin B=5/13

To find The value of tan(A+B)

Solution: We know that sin²A+cos²A=1

                                      sin²B+cos²B=1

∴ sin²A=1-cos²A

⇒ sin A= √(1-cos²A)

⇒ sin A= √(1-(3/5)²)

             = √(1-(9/25))

             = √((25-9)/25)

             =√16/25

             = 4/5

Similarly, os²B=1-sin²B

⇒ cos B=√(1-sin²B)

⇒ cos B= √(1-(5/13)²)

             = √(1-(25/169))

             = √((169-25)/169)

             = √144/169

             = 12/13

We know that tan A=sin A/cos A

                       tan B=sin B/cos B

∴ tan A= (4/5)/(3/5)  [substituting the value of sin A and cos A]

           =(4×5)/(3×5)

           =4/3

tan B= (5/13)/(12/13)  [substituting the value of sin B and cos B]

        = (5×13)/(12×13)

        = 5/12

From the formula of tan(A+B), we have

tan(A+B)= tan A+tan B/(1-(tan A×tan B))

             = {(4/3)+(5/12)}/(1-((4/3)×(5/12)))

             ={((4×4)+5)/12}/(1-(5/9))  [∵L.C.M of 3,12 is 12]

             ={(16+5)/12}/((9-5)/9)

             =(21/12)/(4/9)

             =(21×9)/(4×12)

             =189/48

             = 63/16

Hence the value of tan(A+B) is 63/16.