Question

find m∠c. round to the nearest degree.

Explanation:

Step1: Identify the Law to Use

We have a triangle with sides \( BC = 8 \), \( AC = 12 \), and \( AB = 16 \). To find \( \sin\angle C \), we can first use the Law of Cosines to find \( \cos\angle C \), then use the Pythagorean identity \( \sin^2\theta+\cos^2\theta = 1 \) to find \( \sin\angle C \).

The Law of Cosines states that for a triangle with sides \( a \), \( b \), \( c \) opposite angles \( A \), \( B \), \( C \) respectively, \( c^{2}=a^{2}+b^{2}-2ab\cos C \). In triangle \( ABC \), for angle \( C \), the sides adjacent to \( \angle C \) are \( BC = 8 \) and \( AC = 12 \), and the side opposite is \( AB = 16 \). So we have:

\( AB^{2}=BC^{2}+AC^{2}-2\cdot BC\cdot AC\cdot\cos\angle C \)

Substituting the values \( AB = 16 \), \( BC = 8 \), \( AC = 12 \):

\( 16^{2}=8^{2}+12^{2}-2\times8\times12\times\cos\angle C \)

Step2: Solve for \( \cos\angle C \)

First, calculate the squares:

\( 256 = 64 + 144-192\cos\angle C \)

Simplify the right - hand side:

\( 256=208 - 192\cos\angle C \)

Subtract 208 from both sides:

\( 256 - 208=-192\cos\angle C \)

\( 48=-192\cos\angle C \)

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

\( \cos\angle C=\frac{48}{-192}=-\frac{1}{4}=- 0.25 \)

Step3: Solve for \( \sin\angle C \)

Using the Pythagorean identity \( \sin^{2}\angle C+\cos^{2}\angle C = 1 \), we can solve for \( \sin\angle C \):

\( \sin\angle C=\sqrt{1-\cos^{2}\angle C} \) (since \( \angle C \) is an angle in a triangle, \( 0<\angle C<180^{\circ} \), and \( \sin\) is positive in this interval)

Substitute \( \cos\angle C=- 0.25 \):

\( \sin\angle C=\sqrt{1 - (-0.25)^{2}}=\sqrt{1 - 0.0625}=\sqrt{0.9375}\approx0.9682 \)

Step4: Find the Angle (Optional for \( \sin\), but if we want to check)

If we want to find the angle \( \angle C \) whose sine is approximately \( 0.9682 \), we use the inverse sine function:

\( \angle C=\arcsin(0.9682)\approx75.5^{\circ}\approx76^{\circ} \), but we were asked for \( \sin\angle C \), so we can just present the value of \( \sin\angle C \)

Answer:

\( \sin\angle C\approx0.97 \) (or if we consider the angle first and then sine, but the main value of \( \sin\angle C\) is approximately \( 0.97 \))