Statements AABC with altitude sin(A) C c. sin(A) = h h BD and cos(4) r C Reasons Given Reason 1 Multiplication property of equality c. cos(A) = r a² = 1² + (b − 1)² a = (sin(A))² + (b-c: cos(A)) ² Reason 2 Substitution a² = c² sin²(A) + [b² − 2bc · cos(A) + cos²(A)] Expand by multiplication 2 a=²[sin²(A) + cos² (A)] + b² − 2bc.cos(A) Reason 3 sin²(A) + cos² (A) = 1 a = [1] + b²-2bc cos(A) +6²-2bc cos(A) Reason 4 Substitution Reason 5 =
Statements AABC with altitude sin(A) C c. sin(A) = h h BD and cos(4) r C Reasons Given Reason 1 Multiplication property of equality c. cos(A) = r a² = 1² + (b − 1)² a = (sin(A))² + (b-c: cos(A)) ² Reason 2 Substitution a² = c² sin²(A) + [b² − 2bc · cos(A) + cos²(A)] Expand by multiplication 2 a=²[sin²(A) + cos² (A)] + b² − 2bc.cos(A) Reason 3 sin²(A) + cos² (A) = 1 a = [1] + b²-2bc cos(A) +6²-2bc cos(A) Reason 4 Substitution Reason 5 =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.3: Trigonometric Functions Of Real Numbers
Problem 38E
Related questions
Question
Transcribed Image Text:Complete the proof of the derivation of the law of cosines from the Pythagorean theorem by
providing the missing reasons.
Given: AABC with altitude BD
Prove: a² = b² + c² - 2bc cos(A)
A
C
r
b
h
B
D
a
C
![Statements
AABC with altitude BD
h
= = and cos(4)
- -
sin(A) =
с
c. sin(A) = h
=
r
C
Reasons
Given
Reason 1
Multiplication property of
equality
c· cos(A) = r
a² = 1² + (b − 1)²2²
-
a = (c.sin(A))² + (b − c · cos(A))²
Reason 2
Substitution
a² = c²₂. sin²(A) + [b² − 2bc · cos(A) + COS²(A)] Expand by multiplication
cos
a = c²[sin²(A) + cos²(A)] + b²-2bc cos(A) Reason 3
·
sin² (A) + cos² (A) = 1
a² = c²[1] + b² = 2bc · cos(A)
Reason 4
Substitution
Reason 5
a² = c²2²² +6²-2bc cos(A)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2df48705-4238-4207-b57b-19ed7b40517e%2Fa695727e-76ef-4467-a3d9-1a0009b05307%2Fnprokya_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Statements
AABC with altitude BD
h
= = and cos(4)
- -
sin(A) =
с
c. sin(A) = h
=
r
C
Reasons
Given
Reason 1
Multiplication property of
equality
c· cos(A) = r
a² = 1² + (b − 1)²2²
-
a = (c.sin(A))² + (b − c · cos(A))²
Reason 2
Substitution
a² = c²₂. sin²(A) + [b² − 2bc · cos(A) + COS²(A)] Expand by multiplication
cos
a = c²[sin²(A) + cos²(A)] + b²-2bc cos(A) Reason 3
·
sin² (A) + cos² (A) = 1
a² = c²[1] + b² = 2bc · cos(A)
Reason 4
Substitution
Reason 5
a² = c²2²² +6²-2bc cos(A)
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