Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Solve each triangle. \(a=28\) feet, \(b=47\) feet, \(c=58\) feet

Step-by-Step Solution

Answer

A triangle with sides 28, 47, and 58 has angles \(A \approx 82.2^\circ\), \(B \approx 36.94^\circ\), and \(C \approx 60.86^\circ\).

Step 1: Check Triangle Validity

To ensure the three sides can form a triangle, verify the triangle inequality theorem: \1. \(a + b > c\), which is \(28 + 47 > 58\), so \(75 > 58\) is true. \2. \(a + c > b\), which is \(28 + 58 > 47\), so \(86 > 47\) is true. \3. \(b + c > a\), which is \(47 + 58 > 28\), so \(105 > 28\) is true. \Since all conditions are satisfied, these sides do form a triangle.

Step 2: Use the Law of Cosines to Find an Angle

To find one angle, use the law of cosines. Solve for angle \(A\): \\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(A) \] \\[ 58^2 = 28^2 + 47^2 - 2 \cdot 28 \cdot 47 \cdot \cos(A) \] \\[ 3364 = 784 + 2209 - 2632 \cdot \cos(A) \] \\[ 3364 = 2993 - 2632 \cdot \cos(A) \] \\[ 371 = 2632 \cdot \cos(A) \] \\[ \cos(A) = \frac{371}{2632} \] \\[ A = \cos^{-1}\left(\frac{371}{2632}\right) \] \Calculate \(A\) to find the angle in degrees.

Step 3: Calculate Angle A

Use a calculator to find \(A\): \\[ A \approx \cos^{-1}\left(\frac{371}{2632}\right) \approx 82.2^\circ \] \The calculated angle \(A\) is approximately \(82.2^\circ\).

Step 4: Use the Law of Cosines to Find Another Angle

Now find angle \(B\) using the law of cosines: \\[ b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \] \\[ 47^2 = 28^2 + 58^2 - 2 \cdot 28 \cdot 58 \cdot \cos(B) \] \\[ 2209 = 784 + 3364 - 3248 \cdot \cos(B) \] \\[ 2209 = 4148 - 3248 \cdot \cos(B) \] \\[ -1939 = -3248 \cdot \cos(B) \] \\[ \cos(B) = \frac{1939}{3248} \] \\[ B = \cos^{-1}\left(\frac{1939}{3248}\right) \] \Calculate \(B\) to find the angle in degrees.

Step 5: Calculate Angle B

Use a calculator to find \(B\): \\[ B \approx \cos^{-1}\left(\frac{1939}{3248}\right) \approx 36.94^\circ \] \The calculated angle \(B\) is approximately \(36.94^\circ\).

Step 6: Calculate the Last Angle C

To find angle \(C\), use the fact that the sum of angles in a triangle is \(180^\circ\): \\[ C = 180^\circ - A - B \] \\[ C = 180^\circ - 82.2^\circ - 36.94^\circ \] \Calculate \(C\).

Step 7: Final Calculation of Angle C

Perform the subtraction to find \(C\): \\[ C = 180^\circ - 82.2^\circ - 36.94^\circ = 60.86^\circ \] \So, angle \(C\) is approximately \(60.86^\circ\).

Key Concepts

Triangle Inequality Theorem

The Triangle Inequality Theorem is a crucial concept in triangle geometry. It states that, for any three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This ensures that the sides can indeed "meet" to form a closed shape.

In the given problem, we have sides of lengths 28, 47, and 58 feet. To verify these can make a triangle, we apply the theorem by checking:

• First, ensure that 28 + 47 > 58. This simplifies to 75 > 58, which is true.
• Second, confirm that 28 + 58 > 47. This turns into 86 > 47, which also holds.
• Lastly, ensure 47 + 58 > 28, which simplifies to 105 > 28. Again, this is true.

Thus, the sides satisfy the triangle inequality theorem, affirming they can form a triangle. This theorem is fundamental in examining whether a set of sides can be used to form a valid triangle.

Angle Calculation

Once we confirm a valid triangle, calculating the angles is the next step, and the Law of Cosines is an effective way to start. It relates the sides of a triangle to the cosine of one of its angles, giving:

• For angle A: \[c^2 = a^2 + b^2 - 2ab \cdot \cos(A)\]
• Solving for \(A\): \[\cos(A) = \frac{c^2 - a^2 - b^2}{-2ab}\]
• Substitute the known values (28, 47, 58) to find \(A\).

Using these equations:

• Calculate \(A\) using \(\cos(A) = \frac{371}{2632}\), which results in \(A \approx 82.2^\circ\).
• Find \(B\) similarly, via \(\cos(B) = \frac{1939}{3248}\), resulting in \(B \approx 36.94^\circ\).

The angles are determined with accuracy, thanks to using the cosine values and understanding how they relate to side lengths in non-right angle triangles.

Triangle Geometry

Triangle geometry encompasses the relationships and properties of triangles. One essential property is the angle sum, which always totals 180 degrees in Euclidean triangles.

With two angles found using the Law of Cosines (\(A = 82.2^\circ\) and \(B = 36.94^\circ\)), finding the third angle \(C\) is straightforward:

• Utilize \(C = 180^\circ - A - B\).
• Calculate \(C\): \[C = 180^\circ - 82.2^\circ - 36.94^\circ \approx 60.86^\circ\]

This calculation ensures that all the angles add up to the required 180 degrees, adhering to the basic principles of triangle geometry. Understanding these angle relationships helps deepen comprehension of triangle structures and their properties.