Trigonometric Identities

Let

PM

be lighthouse.

Given,

%5Cangle%20P%20Q%20M%3D%5Calpha%2C%20%5Cangle%20P%20R%20M%3D%5Cbeta

and

Q%20R%3Dd

%5CDelta%20PQM

tan%5C%2C%5Calpha%3D%5Cfrac%7BPM%7D%7BQM%7D

%5CRightarrow%20QM%3DPM%5C%2Ccot%5C%2C%5Calpha

%5CDelta%20PRM

%5Ctan%5C%2C%5Cbeta%3D%5Cfrac%7BPM%7D%7BRM%7D

$\Rightarrow RM = PM\,\cot\,??

%5Ctherefore%20QR%3DQM%2BMR

%5CRightarrow%20d%3DPM%5C%2Ccot%5C%2C%5Calpha%2BPM%5C%2C%20%5Ccot%5C%2C%5Cbeta

%5CRightarrow%20d%3DPM%5Cleft(%5Ccot%5C%2C%5Calpha%20%2B%20%5Ccot%5C%2C%5Cbeta%5Cright)

%5CRightarrow%20d%3DPM%5Cleft(%5Ccot%5C%2C%5Calpha%20%2B%20%5Ccot%5C%2C%5Cbeta%5Cright)

%5CRightarrow%20PM%3D%5Cfrac%7Bd%7D%7B%5Ccot%5C%2C%5Calpha%20%2B%20%5Ccot%5C%2C%5Cbeta%7D

%5Ctherefore

Height of light house is

%5Cfrac%7Bd%7D%7B%5Ccot%5C%2C%5Calpha%20%2B%20%5Ccot%5C%2C%5Cbeta%7D

We have,

%5Cleft(1%2B%5Ctan%201%5E%7B%5Ccirc%7D%5Cright)%5Cleft(1%2B%5Ctan%202%5E%7B%5Ccirc%7D%5Cright)%20%5Cldots%20.%5Cleft(1%2B%5Ctan%2045%5E%7B%5Ccirc%7D%5Cright)%3D2%5E%7B%20n%20%7D

Now,

(1%2B%5Ctan%20%5Ctheta)%5Cleft(1%2B%5Ctan%20%5Cleft(45%5E%7B%5Ccirc%7D-%5Ctheta%5Cright)%5Cright)

%3D(1%2B%5Ctan%20%5Ctheta)%5Cleft(1%2B%5Cfrac%7B1-%5Ctan%20%5Ctheta%7D%7B1%2B%5Ctan%20%5Ctheta%7D%5Cright)

%3D(1%2B%5Ctan%20%5Ctheta)%20%5Cfrac%7B(1%2B%5Ctan%20%5Ctheta%2B1-%5Ctan%20%5Ctheta)%7D%7B1%2B%5Ctan%20%5Ctheta%7D%3D2

%5Ctherefore%5Cleft(1%2B%5Ctan%20%5E%7B%5Ccirc%7D%5Cright)%5Cleft(1%2B%5Ctan%2044%5E%7B%5Ccirc%7D%5Cright)%5Cleft(1%2B%5Ctan%202%5E%7B%5Ccirc%7D%5Cright)%5Cleft(1%2B%5Ctan%2043%5E%7B%5Ccirc%7D%5Cright)

%5Cldots%5Cleft(1%2B%5Ctan%2022%5E%7B%5Ccirc%7D%5Cright)%5Cleft(1%2B%5Ctan%2023%5E%7B%5Ccirc%7D%5Cright)%5Cleft(1%2B%5Ctan%2045%5E%7B%5Ccirc%7D%5Cright)

%3D2%5Ctimes2%5Ctimes2%5Cldots%5Cldots%5Cldots%2022

times

%5Ctimes%20%5Cleft(1%2B1%5Cright)

%3D2%5E%7B22%7D%5Ctimes2%3D2%5E%7B23%7D

%5Ctherefore%20n%3D23

Chapter Questions
3 MCQs