Complete the proof of Heron's formula. 1 a. Begin with the formula Area =bc sin A and square both sides. b. In the equation from part (a), replace sin²A by 1 - cos² A. c. In the equation from part (b), factor 1 - cos² A as a difference of squares. d. Take the square root of both sides and write the area as Area = cos A e. Use the law of cosines to show that a+b+c -a+b+c bc (1+cos A): 2 2 Note: Using similar logic, we can also show that a-b+c a a+b-c bc (1-cos A)= 2 2 f. Use the substitutions = (a+b+c) to rewrite equation (2) as bc (1 bc (1+cos A)s(sa) Note: Using similar logic, we can also show that bc (1-cos A)=(s-b)(s-c) g. Substitute equations (4) and (5) into equation (1). € 2 (3) (4)
Complete the proof of Heron's formula. 1 a. Begin with the formula Area =bc sin A and square both sides. b. In the equation from part (a), replace sin²A by 1 - cos² A. c. In the equation from part (b), factor 1 - cos² A as a difference of squares. d. Take the square root of both sides and write the area as Area = cos A e. Use the law of cosines to show that a+b+c -a+b+c bc (1+cos A): 2 2 Note: Using similar logic, we can also show that a-b+c a a+b-c bc (1-cos A)= 2 2 f. Use the substitutions = (a+b+c) to rewrite equation (2) as bc (1 bc (1+cos A)s(sa) Note: Using similar logic, we can also show that bc (1-cos A)=(s-b)(s-c) g. Substitute equations (4) and (5) into equation (1). € 2 (3) (4)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Complete the proof of Heron's formula.
1
a. Begin with the formula Area =bc sin A and square both sides.
b. In the equation from part (a), replace sin²A by 1 - cos² A.
c. In the equation from part (b), factor 1 - cos² A as a difference of squares.
d. Take the square root of both sides and write the area as
Area =
cos A
e. Use the law of cosines to show that
a+b+c -a+b+c
bc (1+cos A): 2
2
Note: Using similar logic, we can also show that
a-b+c a a+b-c
bc (1-cos A)= 2
2
f. Use the substitutions = (a+b+c) to rewrite equation (2) as
bc (1
bc (1+cos A)s(sa)
Note: Using similar logic, we can also show that
bc (1-cos A)=(s-b)(s-c)
g. Substitute equations (4) and (5) into equation (1).
€
2
(3)
(4)
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