Chapter 7.5, Problem 7AYU

To determine

  $\sin \left(\alpha +\beta \right)=\sin \alpha +\sin \beta +2\sin \alpha \sin \beta$

Answer to Problem 7AYU

  $\sin \left(\alpha +\beta \right)=\sin \beta +\sin \alpha +2\sin \alpha \sin \beta$ is false

Explanation of Solution

Given information:

  $\sin \left(\alpha +\beta \right)=\sin \alpha +\sin \beta +2\sin \alpha \sin \beta$

Calculation:

Consider a circle of unit radius and centre O as shown below

Let us consider point A on the circle on X axis .

Therefore the coordinates of point A will be $\left(1,0\right)$

Let us consider another point B on the circle. Point B lies in first quadrant of circle,

Let $\angle BOA=\beta$

Let us consider another point D on the circle. Point D lies in first quadrant of circle ,

Let $\angle DOB=\alpha$

Let line segment $DF\perp OA$ intersects OA at point D.

Let line segment $DE\perp OB$ intersects OB at point E.

In right $\Delta OED$

  $\begin{array}{l}\frac{OE}{OD}=\cos \alpha \\ OE=1\times \cos \alpha \left[OD=radiusofcircle=1\left(Assumed\right)\right]\end{array}$

  $=\cos \alpha$

  $\begin{array}{l}\frac{DE}{OD}=\sin \alpha \\ DE=1\times \sin \alpha \left[OD=radiusofcircle=1\left(Assumed\right)\right]\end{array}$

  $=\sin \alpha$

Let $EG\perp OA$ intersects OA at point G

In right $\Delta OGE$

  $\begin{array}{l}\frac{EG}{OE}=\sin \beta \\ EG=\cos \alpha \sin \beta \end{array}$

Let line segment $EH\perp DF$ intersects DF at point H

Therefore $EH\perp DF$

In right $\Delta DHE$

  $\begin{array}{l}\frac{DH}{DE}=\cos \beta \\ DH=DE\cos \beta \end{array}$

  = $\sin \alpha \cos \beta$

In right $\Delta OFD$

  $\begin{array}{l}\frac{FD}{OD}=\sin \left(\alpha +\beta \right)\\ \frac{FH+HD}{1}=\sin \left(\alpha +\beta \right)\end{array}$

  $\begin{array}{l}\cos \alpha \sin \beta +\sin \alpha \cos \beta =\sin \left(\alpha +\beta \right)\left[FH=GE=\cos \alpha \sin \beta \right]\end{array}$

Hence $\sin \left(\alpha +\beta \right)=\cos \alpha \sin \beta +\sin \alpha \cos \beta$

Therefore $\sin \left(\alpha +\beta \right)=\sin \beta +\sin \alpha +2\sin \alpha \sin \beta$ is false