Chapter 7: Q. 108 (page 487)

If $α + β + γ = 180^{\circ}$ and $c o t θ = c o t α + c o t β + c o t γ, 0 < θ < 180^{\circ}$
Show that $\sin^{3} θ = \sin (α - θ) \sin (β - θ) \sin (γ - θ)$

Short Answer

Expert verified

If $α + β + γ = 180^{\circ}$ and $c o t θ = c o t α + c o t β + c o t γ, 0 < θ < 180^{\circ}$

Then

$\sin (α - θ) = \frac{\sin^{2} α \sin θ}{\sin β \sin γ} \sin (β - θ) = \frac{\sin^{2} β \sin θ}{\sin α \sin γ} \sin (γ - θ) = \frac{\sin^{2} γ \sin θ}{\sin β \sin α}$

On multiplying all equations

$\sin (α - θ) \sin (β - θ) \sin (γ - θ) = \frac{\sin^{2} α \sin θ}{\sin β \sin γ} \times \frac{\sin^{2} β \sin θ}{\sin α \sin γ} \times \frac{\sin^{2} γ \sin θ}{\sin β \sin α} \sin (α - θ) \sin (β - θ) \sin (γ - θ) = \sin^{2} θ$

Step by step solution

Step 1. Given data

The given equations are

$α + β + γ = 180^{\circ} c o t θ = c o t α + c o t β + c o t γ 0 < θ < 180^{\circ}$

The equation we need to prove is

$\sin^{3} θ = \sin (α - θ) \sin (β - θ) \sin (γ - θ)$

Step 2. Formation of the equation for proof

As $c o t θ = c o t α + c o t β + c o t γ$

$c o t θ = c o t α + c o t β + c o t γ c o t θ - c o t α = c o t β + c o t γ \frac{\cos θ}{\sin θ} - \frac{\cos α}{\sin α} = \frac{\cos β}{\sin β} + \frac{\cos γ}{\sin γ} \frac{\cos θ \sin α - \cos α \sin θ}{\sin θ \sin α} = \frac{\cos β \sin γ + \cos γ \sin β}{\sin β \sin γ} \frac{\sin (α - θ)}{\sin θ \sin α} = \frac{\sin (γ + β)}{\sin β \sin γ} a n d α + β + γ = 180^{\circ} β + γ = 180^{\circ} - α S o \frac{\sin (α - θ)}{\sin θ \sin α} = \frac{\sin (180^{\circ} - α)}{\sin β \sin γ} \frac{\sin (α - θ)}{\sin θ \sin α} = \frac{\sin (α)}{\sin β \sin γ} \sin (α - θ) = \frac{\sin^{2} α \sin θ}{\sin β \sin γ} (i)$

Similarly

$\sin (β - θ) = \frac{\sin^{2} β \sin θ}{\sin α \sin γ} (i i) \sin (γ - θ) = \frac{\sin^{2} γ \sin θ}{\sin β \sin α} (i i i)$

Step 3. proof

Multiply equation(i),(ii),and(iii)

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If $α + β + γ = 180^{\circ}$ and $c o t θ = c o t α + c o t β + c o t γ, 0 < θ < 180^{\circ}$
Show that $\sin^{3} θ = \sin (α - θ) \sin (β - θ) \sin (γ - θ)$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Formation of the equation for proof

Step 3. proof

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If α+β+γ=180∘and cotθ=cotα+cotβ+cotγ,0<θ<180∘Show thatsin3θ=sin(α-θ)sin(β-θ)sin(γ-θ)

Short Answer

Step by step solution

Step 1. Given data

Step 2. Formation of the equation for proof

Step 3. proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

If $α + β + γ = 180^{\circ}$ and $c o t θ = c o t α + c o t β + c o t γ, 0 < θ < 180^{\circ}$
Show that $\sin^{3} θ = \sin (α - θ) \sin (β - θ) \sin (γ - θ)$