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Sagot :
To provide a detailed, step-by-step solution for this question, we need to understand how to develop the law of cosines starting from the definitions of trigonometric functions. As given, you start with sine and cosine definitions related to a right triangle formed by the altitude [tex]\( h \)[/tex].
Here are the steps:
1. Rewriting Trigonometric Equations in Terms of the Numerators:
We know that:
[tex]\[ \sin(A) = \frac{h}{b} \][/tex]
[tex]\[ \cos(A) = \frac{c + r}{b} \][/tex]
Now, solve for [tex]\( h \)[/tex] and [tex]\( c + r \)[/tex]:
[tex]\[ h = b \cdot \sin(A) \][/tex]
[tex]\[ c + r = b \cdot \cos(A) \][/tex]
2. Writing an Expression for [tex]\( r \)[/tex] in Terms of [tex]\( c \)[/tex] and [tex]\( A \)[/tex]:
Since [tex]\( c + r = b \cdot \cos(A) \)[/tex], isolating [tex]\( r \)[/tex]:
[tex]\[ r = b \cdot \cos(A) - c \][/tex]
3. Using Pythagorean Theorem in Both Right Triangles:
Consider the triangle with hypotenuse [tex]\( b \)[/tex]:
[tex]\[ b^2 = h^2 + (c + r)^2 \][/tex]
Substitute [tex]\( h \)[/tex] and [tex]\( c + r \)[/tex] from earlier:
[tex]\[ b^2 = (b \sin(A))^2 + (b \cos(A))^2 = b^2 \sin^2(A) + b^2 \cos^2(A) \][/tex]
4. Combining the Equations:
The identity [tex]\( \sin^2(A) + \cos^2(A) = 1 \)[/tex] gives us:
[tex]\[ b^2 = b^2 (\sin^2(A) + \cos^2(A)) = b^2 \][/tex]
Consider the other triangle with hypotenuse [tex]\( a \)[/tex]:
[tex]\[ a^2 = h^2 + r^2 = (b \sin(A))^2 + (b \cos(A) - c)^2 \][/tex]
Expanding [tex]\( (b \cos(A) - c)^2 \)[/tex]:
[tex]\[ a^2 = b^2 \sin^2(A) + (b \cos(A) - c)^2 \][/tex]
[tex]\[ a^2 = b^2 \sin^2(A) + b^2 \cos^2(A) - 2b \cos(A) c + c^2 \][/tex]
5. Simplifying to Find the Law of Cosines:
Combine [tex]\( b^2 \sin^2(A) + b^2 \cos^2(A) \)[/tex]:
[tex]\[ a^2 = b^2 (\sin^2(A) + \cos^2(A)) - 2bc \cos(A) + c^2 \][/tex]
By using the identity [tex]\( \sin^2(A) + \cos^2(A) = 1 \)[/tex]:
[tex]\[ a^2 = b^2 - 2bc \cos(A) + c^2 \][/tex]
Which gives us the law of cosines:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
Summary of the Fill-in steps:
1. Use the sine ratio to express [tex]\( h \)[/tex] as [tex]\( h = b \sin(A) \)[/tex].
2. Use the cosine ratio to express [tex]\( c + r \)[/tex] as [tex]\( c + r = b \cos(A) \)[/tex].
3. Solve for [tex]\( r \)[/tex] yielding [tex]\( r = b \cos(A) - c \)[/tex].
4. Use the Pythagorean Theorem for both triangles to relate [tex]\( a, b, c, \)[/tex] and [tex]\( A \)[/tex].
5. Combine the information to derive the law of cosines.
Thus, the next drop-down selections would be:
- Rewriting each trigonometric equation in terms of the numerator.
- Carson can write an expression for side [tex]\( r \)[/tex] in terms of [tex]\( b \)[/tex] and [tex]\( c \)[/tex].
- Next, he can use the Pythagorean Theorem to relate [tex]\( a, b, c \)[/tex], and [tex]\( A \)[/tex].
Here are the steps:
1. Rewriting Trigonometric Equations in Terms of the Numerators:
We know that:
[tex]\[ \sin(A) = \frac{h}{b} \][/tex]
[tex]\[ \cos(A) = \frac{c + r}{b} \][/tex]
Now, solve for [tex]\( h \)[/tex] and [tex]\( c + r \)[/tex]:
[tex]\[ h = b \cdot \sin(A) \][/tex]
[tex]\[ c + r = b \cdot \cos(A) \][/tex]
2. Writing an Expression for [tex]\( r \)[/tex] in Terms of [tex]\( c \)[/tex] and [tex]\( A \)[/tex]:
Since [tex]\( c + r = b \cdot \cos(A) \)[/tex], isolating [tex]\( r \)[/tex]:
[tex]\[ r = b \cdot \cos(A) - c \][/tex]
3. Using Pythagorean Theorem in Both Right Triangles:
Consider the triangle with hypotenuse [tex]\( b \)[/tex]:
[tex]\[ b^2 = h^2 + (c + r)^2 \][/tex]
Substitute [tex]\( h \)[/tex] and [tex]\( c + r \)[/tex] from earlier:
[tex]\[ b^2 = (b \sin(A))^2 + (b \cos(A))^2 = b^2 \sin^2(A) + b^2 \cos^2(A) \][/tex]
4. Combining the Equations:
The identity [tex]\( \sin^2(A) + \cos^2(A) = 1 \)[/tex] gives us:
[tex]\[ b^2 = b^2 (\sin^2(A) + \cos^2(A)) = b^2 \][/tex]
Consider the other triangle with hypotenuse [tex]\( a \)[/tex]:
[tex]\[ a^2 = h^2 + r^2 = (b \sin(A))^2 + (b \cos(A) - c)^2 \][/tex]
Expanding [tex]\( (b \cos(A) - c)^2 \)[/tex]:
[tex]\[ a^2 = b^2 \sin^2(A) + (b \cos(A) - c)^2 \][/tex]
[tex]\[ a^2 = b^2 \sin^2(A) + b^2 \cos^2(A) - 2b \cos(A) c + c^2 \][/tex]
5. Simplifying to Find the Law of Cosines:
Combine [tex]\( b^2 \sin^2(A) + b^2 \cos^2(A) \)[/tex]:
[tex]\[ a^2 = b^2 (\sin^2(A) + \cos^2(A)) - 2bc \cos(A) + c^2 \][/tex]
By using the identity [tex]\( \sin^2(A) + \cos^2(A) = 1 \)[/tex]:
[tex]\[ a^2 = b^2 - 2bc \cos(A) + c^2 \][/tex]
Which gives us the law of cosines:
[tex]\[ a^2 = b^2 + c^2 - 2bc \cos(A) \][/tex]
Summary of the Fill-in steps:
1. Use the sine ratio to express [tex]\( h \)[/tex] as [tex]\( h = b \sin(A) \)[/tex].
2. Use the cosine ratio to express [tex]\( c + r \)[/tex] as [tex]\( c + r = b \cos(A) \)[/tex].
3. Solve for [tex]\( r \)[/tex] yielding [tex]\( r = b \cos(A) - c \)[/tex].
4. Use the Pythagorean Theorem for both triangles to relate [tex]\( a, b, c, \)[/tex] and [tex]\( A \)[/tex].
5. Combine the information to derive the law of cosines.
Thus, the next drop-down selections would be:
- Rewriting each trigonometric equation in terms of the numerator.
- Carson can write an expression for side [tex]\( r \)[/tex] in terms of [tex]\( b \)[/tex] and [tex]\( c \)[/tex].
- Next, he can use the Pythagorean Theorem to relate [tex]\( a, b, c \)[/tex], and [tex]\( A \)[/tex].
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