proof of projection rule......
Answers
Answer:
here u go my frnd
Explanation:
The geometrical interpretation of the proof of projection formulae is the length of any side of a triangle is equal to the algebraic sum of the projections of other sides upon it.
In any triangle ABC,
(i) a = b cos C + c cos B
(ii) b = c cos A + a cos C
(iii) c = a cos B + b cos A
Proof:
In any triangle ABC we have a
asinAasinA = bsinBbsinB = csinCcsinC = 2R ……………………. (1)
Now convert the above relation into sides in terms of angles in terms of the sides of any triangle.
a/sin A = 2R
⇒ a = 2R sin A ……………………. (2)
b/sin B = 2R
⇒ b = 2R sin B ……………………. (3)
c/sin c = 2R
⇒ c = 2R sin C ……………………. (4)
(i) a = b cos C + c cos B
Now, b cos C + c cos B
= 2R sin B cos C + 2R sin C cos B
= 2R sin (B + C)
= 2R sin (π - A), [Since, A + B + C = π]
= 2R sin A
= a [From (2)]
Therefore, a = b cos C + c cos B. Proved.
(ii) b = c cos A + a cos C
Now, c cos A + a cos C
= 2R sin C cos A + 2R sin A cos C
= 2R sin (A + C)
= 2R sin (π - B), [Since, A + B + C = π]
= 2R sin B
= b [From (3)]
Therefore, b = c cos A + a cos C.
Therefore, a = b cos C + c cos B. Proved.
(iii) c = a cos B + b cos A
Now, a cos B + b cos A
= 2R sin A cos B + 2R sin B cos A
= 2R sin (A + B)
= 2R sin (π - C), [Since, A + B + C = π]
= 2R sin C
= c [From (4)]
Therefore, c = a cos B + b cos A.
Therefore, a = b cos C + c cos B. Proved
Answer:
Projection law states that in any triangle:
Where A,B,C are the three angled of the triangle and a,b,c are the corresponding opposite side of the angles.
Projection law or the formula of projection law express the algebraic sum of the projection of any two side in term of the third side.
Formula:
a=b cos C+c cos B
b=a cos C+c cos A
c=a cos B+b cos A