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Given: [tex]$\triangle ABC$[/tex] with altitude [tex]$h$[/tex]. Two right triangles are formed: one with side lengths [tex]$c + r, h$[/tex], and [tex]$b$[/tex], and one with side lengths [tex]$r, h$[/tex], and [tex]$a$[/tex].

Carson starts the proof of the law of cosines with [tex]$\sin(A) = \frac{h}{b}$[/tex] by the definition of the sine ratio and [tex]$\cos(A) = \frac{c + r}{b}$[/tex] by the definition of the cosine ratio.

What are the next steps in the proof?

1. Use the [tex]$\square$[/tex] to rewrite each trigonometric equation in terms of the numerator.
2. Then, [tex]$\square$[/tex] [tex]$\square$[/tex] in terms of [tex]$\square$[/tex].
3. Next, he uses the [tex]$\square$[/tex] property of equality to relate [tex]$a, b, c$[/tex], and [tex]$A$[/tex].

Sagot :

GinnyAnsweridn
Here’s a detailed step-by-step solution for completing the proof of the law of cosines using the given information:

1. Rewrite each trigonometric equation in terms of the numerator:

We start with the given trigonometric ratios from the triangle:
[tex]\[ \sin(A) = \frac{h}{b} \][/tex]
[tex]\[ \cos(A) = \frac{c + r}{b} \][/tex]

Rearrange these equations in terms of [tex]\( h \)[/tex] and [tex]\( c + r \)[/tex]:
[tex]\[ h = b \sin(A) \][/tex]
[tex]\[ c + r = b \cos(A) \][/tex]

2. Express [tex]\( r \)[/tex] in terms of the side lengths and angle [tex]\( A \)[/tex]:

Since the triangle with side lengths [tex]\( r, h, a \)[/tex] is a right triangle, use the Pythagorean theorem:
[tex]\[ a^2 = r^2 + h^2 \][/tex]

Substituting [tex]\( h = b \sin(A) \)[/tex] into this equation:
[tex]\[ a^2 = r^2 + (b \sin(A))^2 \][/tex]
[tex]\[ a^2 = r^2 + b^2 \sin^2(A) \][/tex]

3. Use the cosine relationship to find [tex]\( c \)[/tex]:

Recall that [tex]\( c + r = b \cos(A) \)[/tex]. Rearrange this to solve for [tex]\( c \)[/tex]:
[tex]\[ r = b \cos(A) - c \][/tex]

4. Substitute [tex]\( r \)[/tex] back into the equation:

Substitute [tex]\( r = b \cos(A) - c \)[/tex] into the earlier equation [tex]\( a^2 = r^2 + b^2 \sin^2(A) \)[/tex]:
[tex]\[ a^2 = (b \cos(A) - c)^2 + b^2 \sin^2(A) \][/tex]

5. Expand and simplify:

Expand the squared term:
[tex]\[ a^2 = b^2 \cos^2(A) - 2bc \cos(A) + c^2 + b^2 \sin^2(A) \][/tex]

6. Combine like terms:

Recall the Pythagorean identity [tex]\(\cos^2(A) + \sin^2(A) = 1\)[/tex]. Combine the [tex]\( b \)[/tex] terms using this identity:
[tex]\[ a^2 = b^2(\cos^2(A) + \sin^2(A)) - 2bc \cos(A) + c^2 \][/tex]
[tex]\[ a^2 = b^2 - 2bc \cos(A) + c^2 \][/tex]

This is the law of cosines:

[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]

Select the correct answer from each drop-down menu:

1. Use the Pythagorean theorem to rewrite [tex]\(a^2 = r^2 + h^2\)[/tex] in terms of [tex]\(r\)[/tex]:
'a^2 = r^2 + b^2 \sin^2(A)'

2. Then, solve for [tex]\(r\)[/tex] in terms of [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(\cos(A)\)[/tex]:
'r = b \cos(A) - c'

3. And use the substitution property of equality:
'Next, he uses the substitution property of equality to relate [tex]\(a, b, c\)[/tex], and [tex]\(A\)[/tex]'

This guides through steps leading to the law of cosines.
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