Name: Principal Value of ITF: Illustrations 2
Uploaded: 2019-07-19
Duration: 2 min 17 s
Description: Numbers are input and angles are output for inverse trigonometric functions. Let's find the values of missing angles corresponding to given numbers for the c...

Question

Derive Projection formula from Law of $Sines .$

Accepted Answer

We know that law of sines is

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2 R$

$\Rightarrow \frac{a}{\sin A} = 2 R$

$\Rightarrow \frac{b}{\sin B} = 2 R$

$\Rightarrow \frac{c}{\sin C} = 2 R$

$a = 2 R \sin A . . . 1$

$b = 2 R \sin B . . . 2$

$c = 2 R \sin C . . . 3$

Let us take $a = 2 R \sin A . . . 1$

But in triangle sum of angle is $180 °$

$\Rightarrow A + B + C = 180 °$

$\Rightarrow A = 180 ° - (B + C)$

Substitute $A$ in equation $1$

$∴ a = 2 R \sin (180 ° - (B + C))$

$= 2 R \sin (B + C) [∵ \sin (180 - θ) = \sin θ]$

$= 2 R [\sin B \cos C + \cos B \sin C] [∵ \sin (x + y) = \sin x \cos y + \cos x \sin y]$

$= 2 Rsin B \cos C + 2 R \cos B \sin C$

$= (2 R \sin B) \cos C + (2 R \sin C) \cos B$

From equation $2, 3$

$\Rightarrow a = b \cos C + c \cos B$

Now Let us take $b = 2 R \sin B$

Sum of angles is $180 °$ in a triangle

$\Rightarrow A + B + C = 180 °$

$\Rightarrow B = 180 ° - (A + C)$

So, $b = 2 R \sin (180 ° - (A + C))$

$= 2 R \sin (A + C) [∵ \sin (180 ° - θ) = \sin θ]$

$= 2 R [\sin A \cos C + \cos A \sin C] [∵ \sin (x + y) = \sin x \cos y + \cos x \sin y]$

$= (2 R \sin A) \cos C + (2 R \sin C) \cos A$

Substitute equation $1, 2$

$\Rightarrow b = a \cos C + ccos A$

Let us take $c = 2 R \sin C$

Sum of angles of triangle is $180 °$

So $A + B + C = 180 °$

$\Rightarrow C = 180 ° - (A + B)$

$∴ c = 2 R \sin (180 ° - (A + B))$

$= 2 R \sin (A + B) [∵ \sin (180 ° - θ) = \sin θ]$

$= 2 R (\sin A \cos B + \cos A \sin B)$

$= (2 R \sin A) \cos B + (2 R \sin B) \cos A$

Substitute $a = 2 R \sin A and b = 2 R \sin B$

$\Rightarrow c = a \cos B + b \cos A$

Projection formulas from law of sines is

$∴ a = b \cos C + c \cos B$

$b = c \cos A + a \cos C$

$c = a \cos B + b \cos A .$

Derive Projection formula from Law of $Sines .$

Important Questions on Trigonometry

Learn and Prepare for any exam you want !

Derive Projection formula from Law of Sines.

Important Questions on Trigonometry

Learn and Prepare for any exam you want !

Derive Projection formula from Law of $Sines .$