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If an angle be divided into two parts such that the tangent o

Question

If an angle $θ$ be divided into two parts such that the tangent of one part is $m$ times the tangent of the other, then their difference $ϕ$ is given by

cos ϕ = \frac{m - 1}{m + 1} cos θ

sin ϕ = \frac{m - 1}{m + 1} sin θ

sin ϕ = \frac{m - 1}{m + 1} cos θ

cos ϕ = \frac{m - 1}{m + 1} sin θ

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Michael Discussed

If an angle

θ

be divided into two parts such that the tangent of one part is

m

times the tangent of the other, then their difference

ϕ

is given by

9 mins ago

Discuss this question LIVE

9 mins ago

Text solutionVerified

Let the two parts be

A

and

B

, then

A + B = θ, A - B = ϕ

and

tan A = m tan B

\Rightarrow \frac{tan A}{tan B} = \frac{m}{1} \Rightarrow \frac{tan A - tan B}{tan A + tan B} = \frac{m - 1}{m + 1}

(applying componendo and dividendo)

\Rightarrow \frac{sin ( A - B )}{sin ( A + B )} = \frac{m - 1}{m + 1} \Rightarrow sin ϕ = \frac{m - 1}{m + 1} sin θ

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Question Text	If an angle $θ$ be divided into two parts such that the tangent of one part is $m$ times the tangent of the other, then their difference $ϕ$ is given by
Answer Type	Text solution:1
Upvotes	150