9. Bobby was proving the law of cosines and reached a point where he had the equation [tex]b^2 = c^2 \sin^2(B) + c^2 \cos^2(B) + a^2 - 2ac \cos(B)[/tex].

Which of the following steps are needed to complete the proof? Select all that apply.

A. Use the trigonometric identity [tex]\sin^2(B) + \cos^2(B) = 1[/tex] for [tex]b^2 = c^2 + a^2 - 2ac \cos(B)[/tex].

B. Factor [tex]c^2[/tex] for [tex]b^2 = c^2[\sin^2(B) + \cos^2(B)] + a^2 - 2ac \cos(B)[/tex].

C. Use the Pythagorean theorem to result in [tex]b^2 = h^2 + (a - x)^2[/tex].

D. FOIL for [tex]b^2 = h^2 + a^2 - 2ax + x^2[/tex].

Question

17-07-2024
Mathematics

Answered

9. Bobby was proving the law of cosines and reached a point where he had the equation [tex]b^2 = c^2 \sin^2(B) + c^2 \cos^2(B) + a^2 - 2ac \cos(B)[/tex].

Which of the following steps are needed to complete the proof? Select all that apply.

A. Use the trigonometric identity [tex]\sin^2(B) + \cos^2(B) = 1[/tex] for [tex]b^2 = c^2 + a^2 - 2ac \cos(B)[/tex].

B. Factor [tex]c^2[/tex] for [tex]b^2 = c^2[\sin^2(B) + \cos^2(B)] + a^2 - 2ac \cos(B)[/tex].

C. Use the Pythagorean theorem to result in [tex]b^2 = h^2 + (a - x)^2[/tex].

D. FOIL for [tex]b^2 = h^2 + a^2 - 2ax + x^2[/tex].

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-07-17T22:33:35-04:00

To complete the proof that Bobby was working on, let's go through the steps in detail.

### Starting Equation
Bobby has reached the equation:
[tex]\[ b^2 = c^2 \sin^2(B) + c^2 \cos^2(B) + a^2 - 2ac \cos(B) \][/tex]

### Step 1: Factor out [tex]\( c^2 \)[/tex]
First, notice that in the terms [tex]\( c^2 \sin^2(B) \)[/tex] and [tex]\( c^2 \cos^2(B) \)[/tex], we can factor out [tex]\( c^2 \)[/tex]:
[tex]\[ b^2 = c^2 [ \sin^2(B) + \cos^2(B) ] + a^2 - 2ac \cos(B) \][/tex]

### Step 2: Use the Trigonometric Identity
Next, apply the trigonometric identity [tex]\( \sin^2(B) + \cos^2(B) = 1 \)[/tex] to simplify the expression inside the brackets:
[tex]\[ b^2 = c^2 \cdot 1 + a^2 - 2ac \cos(B) \][/tex]
[tex]\[ b^2 = c^2 + a^2 - 2ac \cos(B) \][/tex]

### Conclusion
Thus, the proof steps required are:

- Factor [tex]\( c^2 \)[/tex] for [tex]\( b^2 = c^2 [ \sin^2(B) + \cos^2(B) ] + a^2 - 2ac \cos(B) \)[/tex]
- Use the trigonometric identity [tex]\( \sin^2(B) + \cos^2(B)=1 \)[/tex] for [tex]\( b^2 = c^2 + a^2 - 2ac \cos(B) \)[/tex]

The other given options (use the Pythagorean theorem and FOIL) are not relevant to this proof. Only the steps involving factoring [tex]\( c^2 \)[/tex] and then using the trigonometric identity are necessary to complete Bobby's proof of the law of cosines.