Solving Triangle Problems using the Law of Sines

This question asks us to find the measure of angle B in a triangle ABC, given the lengths of sides a and b, and the sine of angle A. We are provided with the following information:

• Side a = 2
• Side b = 3
• sin A = 2/3

To solve this, we can use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. The formula for the Law of Sines is:

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

We are interested in finding angle B, and we have information about a, b, and sin A. Therefore, we can use the part of the Law of Sines that relates sides a and b to angles A and B:

$$\frac{a}{\sin A} = \frac{b}{\sin B}$$

Now, let's substitute the given values into this equation:

$$\frac{2}{2/3} = \frac{3}{\sin B}$$

To find sin B, we can rearrange the equation. First, simplify the left side:

$$2 \div \frac{2}{3} = 2 \times \frac{3}{2} = 3$$

So the equation becomes:

$$3 = \frac{3}{\sin B}$$

Now, solve for sin B:

$$\sin B = \frac{3}{3}$$

$$\sin B = 1$$

We need to find the angle B whose sine is 1. We know from basic trigonometry that the sine of an angle is 1 at 90 degrees or \(\pi/2\) radians.

Therefore, angle B is equal to \(\pi/2\) radians.

$$B = \arcsin(1) = \frac{\pi}{2}$$

Let's verify if a triangle with these properties is possible. If B = \(\pi/2\), the triangle is a right-angled triangle. In a right triangle, the largest side is opposite the right angle. Here, side b=3 is opposite angle B. Side a=2 is opposite angle A. Since 3 > 2, it is possible for b to be opposite a larger angle than a. Also, the sine of any angle in a triangle must be between 0 and 1 (exclusive for non-degenerate triangles, inclusive for right angles). sin A = 2/3 is between 0 and 1, and sin B = 1 is also a valid value for a triangle angle (specifically a right angle).

Thus, the value of angle B is \(\pi/2\).

Revision Table: Key Information Recap

Additional Information: The Law of Sines and Triangle Properties

The Law of Sines is a fundamental rule in trigonometry used to solve triangles, especially when we know:

• Two angles and one side (AAS or ASA)
• Two sides and a non-included angle (SSA - the ambiguous case)

In this problem, we had the SSA case, but finding sin B = 1 leads to a unique solution (a right triangle), avoiding the ambiguity often associated with SSA.

Key properties of triangles relevant here:

• The sum of angles in a triangle is always \(\pi\) radians or 180 degrees (A + B + C = \(\pi\)).
• The largest side is always opposite the largest angle, and the smallest side opposite the smallest angle.
• The sine of an angle in a triangle is always positive. For angles between 0 and \(\pi\), \(\sin \theta\) is positive.
• If \(\sin B = 1\), then B must be \(\pi/2\) (90 degrees), which means the triangle is a right-angled triangle at vertex B.

Using the Law of Sines is a powerful technique for finding unknown sides or angles in many triangle scenarios.