Problem 16 - Algebra 2

Question

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(A=25^{\circ}, B=78^{\circ}, a=13.7\)

Step-by-Step Solution

Answer

Use the Law of Sines; side b ≈ 31.6, c ≈ 32.0, angle C = 77°.

Step 1: Determine the Law to Use

Evaluate the given information: two angles (A and B) and one side (a) are provided. Use the Law of Sines because we can apply the angle-side-angle (ASA) rule, which is a typical case for using the Law of Sines.

Step 2: Find the Missing Angle C

Using the angle sum property of triangles, calculate the missing angle C. \[ C = 180^\circ - A - B = 180^\circ - 25^\circ - 78^\circ = 77^\circ \]

Step 3: Apply Law of Sines to Find Side b

Use the Law of Sines to find side b. \[\frac{a}{\sin A} = \frac{b}{\sin B} \Rightarrow b = a \cdot \frac{\sin B}{\sin A}\]Substitute known values:\[b = 13.7 \cdot \frac{\sin 78^\circ}{\sin 25^\circ} \approx 31.6\]

Step 4: Apply Law of Sines to Find Side c

Again, using the Law of Sines to find side c. \[\frac{a}{\sin A} = \frac{c}{\sin C} \Rightarrow c = a \cdot \frac{\sin C}{\sin A}\]Substitute known values:\[c = 13.7 \cdot \frac{\sin 77^\circ}{\sin 25^\circ} \approx 32.0\]

Step 5: Summary of Triangle Solutions

The triangle's side lengths are approximately: \(a = 13.7\), \(b = 31.6\), \(c = 32.0\). Angles are: \(A = 25^\circ\), \(B = 78^\circ\), \(C = 77^\circ\).

Key Concepts

Angle-Side-Angle (ASA)

In triangle problems, knowing when to use certain trigonometric laws is essential. The Angle-Side-Angle (ASA) criterion is a useful method for determining the applicability of the Law of Sines. The ASA condition occurs when you know two angles and the side between them. It helps simplify the process of solving a triangle. The law of sines is typically applied when you have this configuration, as it allows for the quick calculation of the unknown opposite sides. In our exercise, we were given angles A and B, and side a. This perfectly fits the ASA condition. This means we can start with the law of sines. To identify ASA in practice, look for:

• Two angles provided in the problem.
• One side known, positioned between the two angles.

Using ASA simplifies solving through the law of sines rather than more complex methods like the law of cosines.

Triangle Angle Sum Property

A fundamental property of triangles is that the sum of their interior angles always equals 180 degrees. This property is often referred to as the Triangle Angle Sum Property. Whenever you know two angles, the third angle can be easily calculated by subtracting the sum of the known angles from 180 degrees. This method ensures that you can fully determine all angles in a triangle if at least two are known. In our given problem, angles A and B were provided. By using the angle sum property, we found that angle C must be 77 degrees. Here's a quick recap of the formula:

• If angles A and B are known, calculate C using: \[ C = 180^ \circ - A - B \]

This makes the process of determining all angles straightforward and guarantees a more comprehensive understanding of triangle geometry.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between side lengths and angles of triangles. One of its powerful tools is the Law of Sines, which is tremendously helpful in solving triangle problems. The Law of Sines states that for any triangle, the ratio of a side length to the sine of its opposite angle is constant for all three sides and angles. The formula is:

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

This formula is indispensable when the ASA or AAS (Angle-Angle-Side) conditions are present, as in our exercise. It allows us to find unknown sides of a triangle when we know two angles and one side, or two sides and one angle. In the exercise, it was used to determine the lengths of sides b and c by rearranging and solving for the unknowns. For instance, to find side b, we used the formula to solve:

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

This calculation simplifies solving triangles and makes understanding their properties much easier.