Problem 25 - Algebra and Trigonometry

Question

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. $$ a=26, \quad c=15, \quad \angle C=29^{\circ} $$

Step-by-Step Solution

Answer

There are two possible triangles: one with angles \( 57.1^{\circ}, 93.9^{\circ}, 29^{\circ} \) and sides \( 26, 30.9, 15 \); the other with angles \( 122.9^{\circ}, 28.1^{\circ}, 29^{\circ} \) and sides \( 26, 12.9, 15 \).

Step 1: Identify Given Values and Setup Equation

We are given \( a = 26 \), \( c = 15 \), and \( \angle C = 29^{\circ} \). We will use the Law of Sines to find \( \angle A \). The Law of Sines states: \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Rearrange this to solve for \( \sin A \): \[ \sin A = \frac{a \cdot \sin C}{c}. \] Substitute the known values: \[ \sin A = \frac{26 \cdot \sin(29^{\circ})}{15}. \]

Step 2: Calculate \( \sin A \)

Calculate \( \sin(29^{\circ}) \) and then solve for \( \sin A \): \[ \sin A = \frac{26 \cdot 0.4848}{15} = 0.8393. \]

Step 3: Determine Possible Angles for \( A \)

Since \( \sin A \approx 0.8393 \), \( \angle A \) can be computed as \( A = \sin^{-1}(0.8393) \approx 57.1^{\circ} \). The ambiguous case suggests that \( \angle A \) could also be \( 180^{\circ} - 57.1^{\circ} = 122.9^{\circ} \). Both solutions need to be checked to ensure they produce valid triangles.

Step 4: Find Possible \( \angle B \) and Check Validity

Using \( \angle A = 57.1^{\circ} \), calculate \( \angle B = 180^{\circ} - 57.1^{\circ} - 29^{\circ} = 93.9^{\circ} \). All angles are positive and less than 180°. Now, check the second case with \( \angle A = 122.9^{\circ} \): \( \angle B = 180^{\circ} - 122.9^{\circ} - 29^{\circ} = 28.1^{\circ} \). Additionally, all angles are positive and less than 180°.

Step 5: Solve for Side \( b \) in Both Cases

Apply the Law of Sines to find \( b \) for both valid triangle solutions. For \( \angle A = 57.1^{\circ} \), \[ b = \frac{a \cdot \sin B}{\sin A} = \frac{26 \cdot \sin(93.9^{\circ})}{\sin(57.1^{\circ})} \approx 30.9. \] For \( \angle A = 122.9^{\circ} \), \[ b = \frac{a \cdot \sin B}{\sin A} = \frac{26 \cdot \sin(28.1^{\circ})}{\sin(122.9^{\circ})} \approx 12.9. \]

Key Concepts

Triangle Solutions

In trigonometry, solving a triangle means finding all its angles and sides. For some triangles, this might involve using the Law of Sines, especially when dealing with non-right triangles. This law is particularly useful when you have certain known values, such as two sides and one angle (SSA scenario), or two angles and one side (AAS or ASA scenarios). In this context, solving a triangle refers to:

• Determining all the missing angles and sides.
• Identifying any possible solutions and ensuring their validity.
• Making use of trigonometric identities to find the values efficiently.

Given values, such as angles and sides, guide the process. You use known angles and sides in the Law of Sines to explore possible triangle configurations that fit the provided data. Once the equations are set up, the sin values and angles can be calculated with precision tools or a scientific calculator.

Angle Calculation

Calculating angles in a triangle involves using trigonometric principles, particularly the Law of Sines or Law of Cosines, depending on what information is available. When applying the Law of Sines:

• The formula is used: \( \frac{a}{\sin A} = \frac{c}{\sin C} \).
• To isolate \( \sin A \), rearrange to find the needed value: \( \sin A = \frac{a \cdot \sin C}{c} \).
• Use a calculator to find \( \sin C \) when given \( \angle C \).

Once you determine \( \sin A \), you can find the angle itself using the inverse sine function (\( \sin^{-1} \)), ensuring the use of degrees if needed. After finding one angle, the remaining angles can be calculated by subtracting from 180°, given the property that the internal angles sum to 180° in any triangle.

Ambiguous Case

The ambiguous case arises when we deal with the SSA configuration of a triangle, where two sides and a non-included angle are known. This situation can potentially lead to:

• No solution.
• One valid triangle.
• Two possible triangles.

For instance, if \( \sin A \) is calculated to be a particular value, \( \angle A \) might have two possible values: \( A \) and \( 180^{\circ} - A \). Both must be checked to see if they lead to valid triangles. Valid triangles satisfy:

• All angles being positive.
• Their sums equaling 180°.
• All sides having positive lengths calculated from the Law of Sines.

When solving using the Law of Sines and encountering this ambiguity, verifying each potential configuration ensures that no solutions are overlooked or incorrectly assumed.

Trigonometry

Trigonometry provides tools like the Law of Sines to solve triangles, especially useful for non-right angled situations. It centers around the relationships between angles and sides of triangles.

• The primary functions are Sine, Cosine, and Tangent, crucial in relating angles to side ratios.
• For solving non-right triangles, the Law of Sines helps compute unknown angles and sides when specific parts are given.
• This law is represented as \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).

In practical terms, trigonometry allows for the prediction of triangle dimensions from known elements. By solving the equalities, students can understand how side lengths and angles coalesce within triangle structures, revealing insights about shape properties in geometric spaces.