Problem 38 - Algebra and Trigonometry

Question

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ b=4, \quad c=5, \quad B=40^{\circ} $$

Step-by-Step Solution

Answer

One triangle exists with sides a ≈ 6.22, b = 4, c = 5 and angles A ≈ 86.9°, B = 40°, C ≈ 53.1°.

Step 1: Determine the possible number of triangles

Given: side b = 4, side c = 5, and angle B = 40°. We need to check if the given information satisfies the conditions for forming a triangle using the Law of Sines. Compute \ \( \frac{b}{\text{sin}(B)} = \frac{4}{\text{sin}(40^\text{°})} \) \ Calculate the ratio: \ \( \frac{4}{\text{sin}(40^\text{°})} = 6.23 \). Now check if this is greater than the given side c = 5.

Step 2: Check if a solution exists

Since \ \(\frac{4}{\text{sin}(40^\text{°})} = 6.23 > 5 \), it means the given sides and angle B can form a triangle. Since the calculated value is greater than the opposite side length, only one triangle will be formed.

Step 3: Use the Law of Sines to find angle C

Using the Law of Sines: \ \( \frac{\text{sin}(C)}{c} = \frac{\text{sin}(B)}{b} \) \ Substitute the values: \ \( \text{sin}(C) = \frac{5 \times \text{sin}(40^\text{°})}{4} \) \ Calculate \ \( \text{sin}(C) = \frac{5 \times 0.6428}{4} = 0.8035 \) \ \( C = \text{sin}^{-1}(0.8035) = 53.1^\text{°} \)

Step 4: Find angle A

Since the sum of angles in a triangle is 180°, we can find angle A: \ \( A = 180^\text{°} - B - C \) \ Substitute the values: \ \( A = 180^\text{°} - 40^\text{°} - 53.1^\text{°} = 86.9^\text{°} \)

Step 5: Use the Law of Sines to find side a

Using the Law of Sines again: \ \( \frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} \) \ Rewrite it to find a: \ \( a = \frac{b \times \text{sin}(A)}{\text{sin}(B)} \) \ Substitute in the values: \ \( a = \frac{4 \times \text{sin}(86.9^\text{°})}{\text{sin}(40^\text{°})} \) \ Calculate the value: \ \( a = \frac{4 \times 0.9995}{0.6428} \) \ \( a = 6.22 \)

Key Concepts

Triangle Existence

Understanding when a triangle can exist is important. For any given triangle, we must satisfy certain conditions. The **Law of Sines** tells us that if we know two sides and the included angle (not between them), we can check if a triangle exists. Given values in our problem are:

• Side b = 4
• Side c = 5
• Angle B = 40°

Using the ratio from the Law of Sines, \(\frac{b}{\text{sin}(B)}\), we compute: \( \frac{4}{\text{sin}(40^\text{°)}} = 6.23 \). Since this value (6.23) is greater than side c (which is 5), a triangle can exist.

Angle-Side Relationship

Once we know a triangle can exist, we explore the relationship between its sides and angles. The **Law of Sines** helps by connecting the given sides and angles. For our triangle, we use the ratio:
\( \frac{\text{sin}(C)}{c} = \frac{\text{sin}(B)}{b} \)
Substituting the known values:
\( \text{sin}(C) = \frac{5 \times \text{sin}(40^\text{°})}{4} \)
This gives:
\( \text{sin}(C) = 0.8035 \)
Which means angle C can be calculated:
\( C = \text{sin}^{-1}(0.8035) = 53.1^\text{°} \).
This angle-side method confirms the integrity of our triangle's shape and size, giving us precise measurements.

Triangle Angles and Sides

With the confirmed angles and sides from previous steps, we can now solve for the remaining unknowns. Knowing the angles in a triangle must sum up to 180° helps us find the last angle, A, using:
\( A = 180^\text{°} - B - C \)
Substituting in the known angles:
\( A = 180^\text{°} - 40^\text{°} - 53.1^\text{°} = 86.9^\text{°} \).
Lastly, to find the missing side a, we use the **Law of Sines** again:
\( \frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} \)
Solving for a, we get:
\( a = \frac{4 \times \text{sin}(86.9^\text{°})}{\text{sin}(40^\text{°})} \)
Resulting in:
\( a = \frac{4 \times 0.9995}{0.6428} = 6.22 \).

• This method ensures we understand how angles and sides interrelate within triangles.