Problem 27 - Algebra 2

Question

Determine whether each triangle has no solution, one solution, or two solutions. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ A=30^{\circ}, a=14, b=28 $$

Step-by-Step Solution

Answer

There is one solution: \( B = 90^{\circ}, C = 60^{\circ}, c \approx 24.2 \).

Step 1: Understand the Given Information

We are given the angle \( A = 30^{\circ} \) and sides \( a = 14 \), \( b = 28 \). We need to determine the number of solutions for the triangle and solve it. This scenario involves the use of the Law of Sines, where we need to check for the ambiguous case by calculating \( \sin B \).

Step 2: Calculate \( \sin B \) Using the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} \). Plugging in the known values, we have: \[ \frac{14}{\sin 30^{\circ}} = \frac{28}{\sin B} \] \( \sin 30^{\circ} \) is \( 0.5 \), thus: \[ \frac{14}{0.5} = \frac{28}{\sin B} \] This simplifies to \( 28 = \frac{28}{\sin B} \). Therefore, \( \sin B = 1 \).

Step 3: Determine the Number of Solutions

Since \( \sin B = 1 \), we must consider if there are valid angles for \( B \) that satisfy this condition. The angle \( B \) with \( \sin B = 1 \) is \( 90^{\circ} \). Thus, this implies there's exactly one solution where \( B = 90^{\circ} \).

Step 4: Solve the Triangle

Since \( B = 90^{\circ} \) and \( A = 30^{\circ} \), we can find angle \( C \) using \( C = 180^{\circ} - A - B = 180^{\circ} - 30^{\circ} - 90^{\circ} = 60^{\circ} \). To find side \( c \), we apply the Law of Sines again: \[ \frac{a}{\sin A} = \frac{c}{\sin C} \] \[ \frac{14}{0.5} = \frac{c}{\sin 60^{\circ}} \] \( \sin 60^{\circ} \) is approximately \( 0.866 \). Thus, \[ 28 = \frac{c}{0.866} \] \[ c = 28 \times 0.866 \approx 24.2 \] So, \( c \approx 24.2 \).

Step 5: Confirm the Solution with Triangle Properties

Check that the angles sum to \( 180^{\circ} \). \( A = 30^{\circ}, B = 90^{\circ}, C = 60^{\circ} \) satisfy this property, confirming our solution.

Key Concepts

Ambiguous Case

In triangle solutions, the ambiguous case arises when you use the Law of Sines to solve a triangle and there might be more than one possible triangle fulfilling the conditions given. This is specifically relevant in situations where you know two sides and a non-included angle, known as the SSA (side-side-angle) condition. These parameters can sometimes create confusion, as they might result in:

• No valid triangle if the given sides and angle don't form a triangle.
• One triangle if the given sides and angle just fit together perfectly.
• Two triangles because the known elements allow for two different configurations.

Always remember that for the ambiguous case, calculating the sine of the unknown angle and considering its possible angles (like the use of the sine rule leading to either acute or obtuse angle solutions, depending on the side length relationships) is crucial. This fundamental understanding helps in identifying conditions where you might face zero, one, or two solutions.

Triangle Solution

Solving a triangle involves finding the unknown angles and sides when certain measurements are known. Utilizing the Law of Sines is common when solving triangles, especially when dealing with the ambiguous case. With given angle and sides, you can determine unknowns using the formula:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]In practical terms, here's how you typically approach the triangle solution process:

• Start with known values and apply the Law of Sines to find one of the unknowns.
• Check your calculated values to see if they are feasible within triangle rules, such as angle summation equaling 180°.
• Continue solving for other unknowns using consistent application of the found data and rules, like cosine or tangent laws if necessary.

This organized approach ensures that each step logically follows from the last, making the solution clearer and more manageable.

Angle Measures

When solving triangles, understanding and determining angles accurately is essential. Each angle in a triangle has characteristics that can help solve the triangle when combined with some known measurements. For instance:

• The sum of all angles in any triangle is always 180°.
• Sine values are linked to specific angles, allowing identification of possible angle measures when sides are known.
• An angle of 90°, classifies the triangle as a right triangle which affects properties, such as one angle always being complementary to another.

Using properties and relationships between angles can simplify solving unknowns. For example, if two angles are known, the third can be found by subtracting their sum from 180°. This fundamental angle measure rule often forms the basis of triangle calculations, simplifying the otherwise complex task of finding side lengths or additional angles.

Sine Function

The sine function is a mathematical function that relates a given angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. It's pivotal in the Law of Sines for triangle calculations. Here's how it works:

• In a triangle, if angle \(A\) is given, the sine function helps find the ratio of side \(a\) to the triangle's hypotenuse.
• The Law of Sines uses sine values to equate ratios of a triangle's sides to its opposite angles, expressed as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
• This function also assists in checking possible angle values (like 0° or 90°) for validity when combined with specific side lengths.

Understanding how the sine function works provides a clear path to using trigonometric principles to solve triangles. It's crucial for identifying specific angle and side measures based directly on initial input through clear mathematical insights.