Question

the law of cosines
follow these steps to derive the law of cosines.
...
c² = k² + h² (using the equation from step 1, the equation ( a^2 = (b - x)^2 + h^2 ) becomes ( a^2 = b^2 - 2bx + x^2 + h^2 ). by substitution...
c. in ( \triangle abd ), the trigonometric function ( cos(angle a) = \frac{x}{c} )
d. multiply both sides of the equation in step b by ( c ) to get ( x = c cos(a) )

1. substitute x for the variable ( x ) in the equation ( a^2 = b^2 - 2bx + x^2 + h^2 ) to produce ( a^2 = b^2 - 2bc cos(a) + c^2 )

(diagram: triangle ( abc ) with right angle at ( d ), ( ad = x ), ( dc = b - x ), ( ab = c ), ( bc = a ), ( ac = b ), ( bd = h ))

Explanation:

To determine the correct substitution for \( x \) in the equation \( a^2 = b^2 - 2bx + x^2 \) (derived from the geometric setup of the triangle and the Pythagorean theorem) to derive the Law of Cosines, we analyze the right triangle \( \triangle ABD \):

Step 1: Recall the trigonometric relationship in \( \triangle ABD \)

In right triangle \( \triangle ABD \), the cosine of angle \( A \) is defined as the adjacent side over the hypotenuse. The adjacent side to angle \( A \) is \( x \), and the hypotenuse is \( c \). So:
\[
\cos(A) = \frac{x}{c}
\]

Step 2: Solve for \( x \)

Multiply both sides of the equation \( \cos(A) = \frac{x}{c} \) by \( c \) to isolate \( x \):
\[
x = c \cdot \cos(A)
\]

Step 3: Substitute \( x = c \cos(A) \) into the equation

When we substitute \( x = c \cos(A) \) into \( a^2 = b^2 - 2bx + x^2 \), we get:
\[
a^2 = b^2 - 2b(c \cos(A)) + (c \cos(A))^2
\]
Simplifying this (and using the Pythagorean identity \( b^2 - (c \cos(A))^2 = (b \sin(A))^2 \), though the key substitution here is for \( x \)) leads to the Law of Cosines.

The correct substitution for \( x \) is \( \boldsymbol{c \cos(A)} \) (from the options, this corresponds to "c cos(A)" in the provided choices).

Answer:

\( c \cos(A) \) (or the option labeled with \( c \cos(A) \) in the original problem's choices)