prove projection rule
b= c cosA + a cosC
c= a cosB. + b cosA
Answers
Answer:
Let ABC be a triangle. Then the following three cases arises:
Case I: If ABC is an acute-angled triangle then we get,
a = BC = BD + CD ………………………… (i)
Now from the triangle ABD we have,
cos B = BD/AB
⇒ BD = AB cos B
⇒ BD = c cos B, [since, AB = c]
Again, cos C = CD/AC
⇒ CD = AC cos C
⇒ CD = b cos C, [since, AC = b]
Projection Formulae
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Now, substitute the value of BD and CD in equation (i) we get,
a = c cos B + b cos C
Note: We observe in the above diagram BD and CD are projections of AB and AC respectively on BC.
Case II: If ABC is an acute-angled triangle then we get,
a = BC = CD - BD ………………………… (ii)
Now from the triangle ADC we have,
cos C = CD/AC
⇒ CD = AC cos C
⇒ CD = b cos C, [since, AC = b]
Again, cos (π - B) = BD/AB
⇒ BD = AB cos (π - B)
a = b cos C + c cos B
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⇒ BD = -c cos B, [since, AB = c and cos (π - θ) = -cos θ]
Now, substitute the value of BD and CD in equation (ii) we get,
a = b cos C - (-c cos B)
⇒ a = b cos C + c cos B
Case III: If ABC is a right-angled triangle then we get,
a = BC ………………………… (iii)
and cos B = BC/AB
⇒ BC = AB cos B
⇒ BC = c cos B, [since, AB = c]
Now, substitute the value of BC in equation (iii) we get,
a = c cos B
⇒ a = c cos B + 0
b = c cos A + a cos C
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⇒ a = c cos B + b cos C, [since C = 90° ⇒ cos C = cos 90 = 0]
Therefore, in any triangle ABC we get, a = b cos C + c cos B
Similarly, we can prove that the formulae b = c cos A + a cos C and c = a cos B + b cos A.
Step-by-step explanation:
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