Let's work through the steps of proving the law of cosines for [tex]\(\triangle ABC\)[/tex]:
1. Given Definitions by Trigonometric Ratios:
- By the definition of the sine ratio: [tex]\(\sin(A) = \frac{h}{b}\)[/tex]
- By the definition of the cosine ratio: [tex]\(\cos(A) = \frac{c + r}{b}\)[/tex]
2. Rewrite Each Trigonometric Equation:
- [tex]\(\sin(A) = \frac{h}{b}\)[/tex]
- [tex]\(\cos(A) = \frac{c + r}{b}\)[/tex]
3. Express in Terms of the Numerator:
- [tex]\(h = b \sin(A)\)[/tex]
- [tex]\(c + r = b \cos(A)\)[/tex]
4. Expression for Side [tex]\(r\)[/tex]:
- From [tex]\(c + r = b \cos(A)\)[/tex], solve for [tex]\(r\)[/tex]:
[tex]\[
r = b \cos(A) - c
\][/tex]
5. Using the Law of Cosines:
- The law of cosines relates [tex]\(a, b, c\)[/tex], and angle [tex]\(A\)[/tex]. According to the law of cosines:
[tex]\[
a^2 = b^2 + c^2 - 2bc \cos(A)
\][/tex]
In summary, the steps in the proof are:
- Use trigonometric definitions to express [tex]\(h\)[/tex] and [tex]\(c + r\)[/tex] in terms of known quantities.
- Solve for [tex]\(r\)[/tex] in terms of [tex]\(b, c\)[/tex], and [tex]\(\cos(A)\)[/tex].
- Apply the law of cosines to relate [tex]\(a, b, c\)[/tex], and angle [tex]\(A\)[/tex].
Correct Answer Selection:
1. Use the trigonometric definitions to rewrite each trigonometric equation in terms of the numerator.
2. Then, Carson can write an expression for side [tex]\(r\)[/tex] in terms of [tex]\(b \cos (A) - c\)[/tex].
3. Next, he can use the law of cosines to relate [tex]\(a, b, c\)[/tex], and angle [tex]\(A\)[/tex].
Therefore, you should select:
1. "trigonometric definitions"
2. "r"
3. "the law of cosines"
