Question

in the triangle shown above, ( a = 16 mathrm{cm}, b = 12 mathrm{cm} ), and ( c = 8 mathrm{cm} ). what is the approximate measure of angle ( a )?

a. ( 65.56^{circ} )
b. ( 75.52^{circ} )
c. ( 104.48^{circ} )
d. ( 14.48^{circ} )

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\).
Substitute \(a = 16\), \(b = 12\), and \(c = 8\) into the formula:
\(16^{2}=12^{2}+8^{2}-2\times12\times8\times\cos A\)
\(256 = 144 + 64-192\cos A\)
\(256=208 - 192\cos A\)

Step2: Solve for \(\cos A\)

Rearrange the equation:
\(192\cos A=208 - 256\)
\(192\cos A=- 48\)
\(\cos A=\frac{-48}{192}=-\frac{1}{4}=- 0.25\)

Step3: Find the angle \(A\)

Use the inverse - cosine function \(A=\cos^{-1}(-0.25)\)
\(A\approx104.48^{\circ}\)

Answer:

C. \(104.48^{\circ}\)