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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?

Use the [tex]$\square$[/tex] to rewrite each trigonometric equation in terms of the numerator.
Then, Carson can write an expression for side [tex]$\square$[/tex] in terms of [tex]$\square$[/tex].
Next, he can use the [tex]$\square$[/tex] to relate [tex]$a, b, c$[/tex], and [tex]$A$[/tex].



Answer :

GinnyAnswer
1. Use Pythagorean theorem to rewrite each trigonometric equation in terms of the numerator.

- [tex]$\sin (A)=\frac{h}{b} \Rightarrow h = b \sin(A)$[/tex]
- [tex]$\cos (A)=\frac{c+r}{b} \Rightarrow c + r = b \cos(A)$[/tex]

2. Then, Carson can write an expression for side [tex]\( r \)[/tex] in terms of [tex]\( c \)[/tex].

- From the second equation, [tex]\( r = b \cos(A) - c \)[/tex]

3. Next, he can use the Pythagorean theorem to relate [tex]\( a, b, c \)[/tex], and [tex]\( A \)[/tex].

- Using the right triangle with side lengths [tex]\( r, h \)[/tex], and [tex]\( a \)[/tex]:
[tex]\[ a^2 = r^2 + h^2 \][/tex]

Substituting [tex]\( r = b \cos(A) - c \)[/tex] and [tex]\( h = b \sin(A) \)[/tex]:
[tex]\[ a^2 = (b \cos(A) - c)^2 + (b \sin(A))^2 \][/tex]
[tex]\[ a^2 = (b \cos(A) - c)^2 + (b \sin(A))^2 \][/tex]
[tex]\[ a^2 = b^2 \cos^2(A) - 2bc \cos(A) + c^2 + b^2 \sin^2(A) \][/tex]
Using the identity [tex]\(\cos^2(A) + \sin^2(A) = 1\)[/tex]:
[tex]\[ a^2 = b^2 (\cos^2(A) + \sin^2(A)) - 2bc \cos(A) + c^2 \][/tex]
[tex]\[ a^2 = b^2 \cdot 1 - 2bc \cos(A) + c^2 \][/tex]
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]

And that's the proof of the law of cosines.

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