Question

find the measure of angle c.
a
c = 8 b = 14
b a = 19 c = ?°
law of cosines: c² = a² + b² - 2ab·cos(c)
round your answer to the nearest hundredth.

Explanation:

Step1: Substitute values into Law of Cosines

Given \( c = 8 \), \( a = 19 \), \( b = 14 \), and the Law of Cosines formula \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \). Substitute the values:
\( 8^2 = 19^2 + 14^2 - 2 \cdot 19 \cdot 14 \cdot \cos(C) \)
Calculate each term:
\( 64 = 361 + 196 - 532 \cdot \cos(C) \)

Step2: Simplify the equation

Combine the constants on the right:
\( 64 = 557 - 532 \cdot \cos(C) \)
Subtract 557 from both sides:
\( 64 - 557 = -532 \cdot \cos(C) \)
\( -493 = -532 \cdot \cos(C) \)

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

Divide both sides by -532:
\( \cos(C) = \frac{-493}{-532} \approx 0.9267 \)

Step4: Find angle \( C \)

Take the inverse cosine:
\( C = \cos^{-1}(0.9267) \approx 21.93^\circ \)

Answer:

\( 21.93 \)