Question

solve for x. law of cosines: $c^2 = a^2 + b^2 - 2ab \cdot \cos(c)$ round your answer to the nearest hundredth. 14, 26, $x$ (angle), 19 $x = ? ^\circ$

Explanation:

Step1: Identify sides for Law of Cosines

In the triangle, the side opposite angle \( x \) is \( 14 \), and the other two sides are \( a = 26 \), \( b = 19 \), \( c = 14 \). The Law of Cosines formula is \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( C = x \). So, \( 14^2 = 26^2 + 19^2 - 2(26)(19)\cos(x) \).

Step2: Simplify the equation

Calculate each term: \( 196 = 676 + 361 - 988\cos(x) \). Combine like terms: \( 196 = 1037 - 988\cos(x) \).

Step3: Solve for \( \cos(x) \)

Subtract \( 1037 \) from both sides: \( 196 - 1037 = -988\cos(x) \), so \( -841 = -988\cos(x) \). Divide both sides by \( -988 \): \( \cos(x) = \frac{841}{988} \approx 0.8512 \).

Step4: Find \( x \) using arccos

Take the arccosine: \( x = \arccos(0.8512) \approx 31.67^\circ \).

Answer:

\( 31.67 \)