Apply the law of sines to the following:


A = 104°, a = 26.8, b = 31.3.


What happens when we try to find the measure of angle B using a calculator?

Verified step by step guidance

1. Step 1: Recall the Law of Sines, which states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
2. Step 2: Substitute the known values into the Law of Sines: \( \frac{26.8}{\sin 104^\circ} = \frac{31.3}{\sin B} \).
3. Step 3: Solve for \( \sin B \) by rearranging the equation: \( \sin B = \frac{31.3 \cdot \sin 104^\circ}{26.8} \).
4. Step 4: Use a calculator to find the value of \( \sin B \).
5. Step 5: Determine if the value of \( \sin B \) is valid (i.e., between -1 and 1) to find angle B. If \( \sin B > 1 \), it indicates an error or impossibility in the triangle configuration.

Key Concepts

Law of Sines

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be expressed as a/sin(A) = b/sin(B) = c/sin(C). It is particularly useful for solving triangles when we know either two angles and one side or two sides and a non-included angle.

Ambiguous Case of the Law of Sines

The ambiguous case occurs when using the Law of Sines to find an angle when two sides and a non-included angle are known (SSA condition). This can lead to zero, one, or two possible solutions for the angle, depending on the given values. Understanding this concept is crucial for determining whether a triangle can be formed with the given measurements.

Calculator Functions for Trigonometry

Calculators typically have functions to compute trigonometric ratios and their inverses, such as sine, cosine, and tangent. When finding an angle using the sine function, the calculator will return the principal value, which may not account for all possible angles in the ambiguous case. Therefore, it is important to analyze the context of the triangle to determine if additional solutions exist.