Question

follow these steps to derive the law of cosines. 2. the relationship between the side lengths in △cbd is ( a^2 = (b - x)^2 + h^2 ) by the pythagorean theorem. 3. the equation ( a^2 = (b - x)^2 + h^2 ) is expanded to become ( a^2 = b^2 - 2bx + x^2 + h^2 ). 4. using the equation from step 1, the equation ( a^2 = b^2 - 2bx + x^2 + h^2 ) becomes ( a^2 = b^2 - 2bx + c^2 ) by substitution. 5. in △abd, the trigonometric function dropdown with cos(a), cos(b), cos(c) ( = \frac{x}{c} ). the diagram shows triangle abc with height ( h ) from b to ac, d is the foot, so ( ad = x ), ( dc = b - x ), ( ab = c ), ( bc = a ), ( ac = b ), with a right angle at d.

Explanation:

Step1: Recall Trigonometric Definitions

In right triangle \( \triangle ABD \), angle \( A \) is at vertex \( A \), adjacent side to \( \angle A \) is \( x \), hypotenuse is \( c \). The cosine of an angle in a right triangle is defined as \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \). For \( \angle A \), adjacent side is \( x \), hypotenuse is \( c \), so \( \cos(A)=\frac{x}{c} \).

Step2: Eliminate Other Options

• \( \cos(B) \): In \( \triangle ABD \), angle \( B \) is not an acute angle of the right triangle in a way that \( \frac{x}{c} \) would correspond to its cosine.
• \( \cos(C) \): Angle \( C \) is in \( \triangle CBD \), not directly related to the ratio \( \frac{x}{c} \) in \( \triangle ABD \).

Answer:

\(\cos(A)\)