In triangle ABC, a = 9.4 m, b = 16.1 m, and angle C = 68 degrees.  Determine angle B to the nearest degree.

To determine angle \(B\) in triangle \(ABC\), we can use the Law of Sines. The Law of Sines states that for any triangle:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

Given:
- \(a = 9.4\) m
- \(b = 16.1\) m
- \(\angle C = 68^\circ\)

We need to find \(\angle B\).

First, use the Law of Sines to express \(\sin B\):

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]

Given that we have \(\angle C\), we can write:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

But we need another relationship to find \(\angle B\). We can use the Law of Cosines to find side \(c\), then use the Law of Sines again.

Using the Law of Cosines to find side \(c\):

\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]

\[ c^2 = 9.4^2 + 16.1^2 - 2 \cdot 9.4 \cdot 16.1 \cdot \cos(68^\circ) \]

Calculate each term:

\[ 9.4^2 = 88.36 \]
\[ 16.1^2 = 259.21 \]
\[ 2 \cdot 9.4 \cdot 16.1 \approx 302.68 \]
\[ \cos(68^\circ) \approx 0.3746 \]

Now substitute these values into the equation:

\[ c^2 = 88.36 + 259.21 - 302.68 \cdot 0.3746 \]
\[ c^2 = 347.57 - 113.43 \]
\[ c^2 = 234.14 \]

\[ c = \sqrt{234.14} \approx 15.3 \]

Now we use the Law of Sines to find \(\angle B\):

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

\[ \frac{9.4}{\sin A} = \frac{16.1}{\sin B} = \frac{15.3}{\sin 68^\circ} \]

First, find \(\sin A\):

\[ \sin A = \frac{9.4 \cdot \sin 68^\circ}{15.3} \]
\[ \sin 68^\circ \approx 0.9272 \]

\[ \sin A = \frac{9.4 \cdot 0.9272}{15.3} \approx 0.5693 \]

Now find \(\angle A\):

\[ A = \arcsin(0.5693) \approx 34.7^\circ \]

Since the sum of the angles in a triangle is \(180^\circ\):

\[ B = 180^\circ - 68^\circ - 34.7^\circ \]
\[ B = 77.3^\circ \]

To the nearest degree, angle \(B\) is \(77^\circ\).

Would you like more details or have any questions?

Here are 5 more questions you can ask:
1. How do you use the Law of Cosines to find an angle given all three sides of a triangle?
2. Can you explain how to find the area of a triangle using the base and height?
3. What are the steps for solving a system of linear equations using the substitution method?
4. How do you determine the critical points of a function?
5. Can you explain the process of evaluating a definite integral using the Fundamental Theorem of Calculus?

Tip: When solving trigonometric equations, double-check your calculator settings to ensure you are using the correct angle units (degrees or radians).

Learn how to determine angle B in triangle ABC with sides a = 9.4 m, b = 16.1 m, and angle C = 68 degrees using the Law of Sines and Law of Cosines. This problem involves trigonometric concepts and is suitable for high school students.

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