Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Solve each triangle. \(a=9.3\) inches, \(b=5.7\) inches, \(c=8.2\) inches

Step-by-Step Solution

Answer

The angles of the triangle are approximately \(81.84^\circ\), \(37.13^\circ\), and \(61.03^\circ\).

Step 1: Verify the Triangle Inequality Theorem

First, check if the given sides can form a triangle by verifying the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides must be greater than the length of the third side.- Check: \( a + b > c \Rightarrow 9.3 + 5.7 = 15 > 8.2 \)- Check: \( a + c > b \Rightarrow 9.3 + 8.2 = 17.5 > 5.7 \)- Check: \( b + c > a \Rightarrow 5.7 + 8.2 = 13.9 > 9.3 \)Since all conditions are satisfied, a triangle can indeed be formed.

Step 2: Use Law of Cosines to Find One Angle

Let's start by finding one angle using the law of cosines, for example \( \angle A \).The law of cosines is given by:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]So, to find \( \angle A \), we use:\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]Substitute the values:\[ \cos(A) = \frac{5.7^2 + 8.2^2 - 9.3^2}{2 \times 5.7 \times 8.2} \]Calculate: \[ \cos(A) = \frac{32.49 + 67.24 - 86.49}{93.48} = \frac{13.24}{93.48} = 0.1417 \]Thus, \( A \approx \arccos(0.1417) \approx 81.84^\circ \).

Step 3: Use Law of Cosines to Find Second Angle

Let's now find \( \angle B \) using the law of cosines:\[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \]Substitute the values:\[ \cos(B) = \frac{9.3^2 + 8.2^2 - 5.7^2}{2 \times 9.3 \times 8.2} \]Calculate: \[ \cos(B) = \frac{86.49 + 67.24 - 32.49}{152.28} = \frac{121.24}{152.28} = 0.7963 \]Thus, \( B \approx \arccos(0.7963) \approx 37.13^\circ \).

Step 4: Find the Remaining Angle

Use the fact that the sum of angles in a triangle is \(180^\circ\) to find \( \angle C \).\[ C = 180^\circ - A - B \]Substitute the known values:\[ C = 180^\circ - 81.84^\circ - 37.13^\circ \]Calculate: \[ C = 61.03^\circ \].

Step 5: Confirm the Triangle's Properties

Finally, re-verify that the triangle follows the fundamental properties:- Check that the angle measures sum to \(180^\circ\): \[ 81.84^\circ + 37.13^\circ + 61.03^\circ = 180^\circ \]- Double-check your math if any computations are off, ensuring all steps and calculations conform to expectations.

Key Concepts

Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem is critical in determining whether three side lengths can form a triangle. It states that 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 connect efficiently to form a triangle rather than a straight line.

In the problem, we have the side lengths \(a = 9.3\), \(b = 5.7\), and \(c = 8.2\). Let's apply the theorem:

• Check if \(a + b > c\): \(9.3 + 5.7 = 15\), which is greater than \(8.2\).
• Check if \(a + c > b\): \(9.3 + 8.2 = 17.5\), again greater than \(5.7\).
• Finally, check if \(b + c > a\): \(5.7 + 8.2 = 13.9\), exceeding \(9.3\).

Since all conditions are satisfied, these side lengths can form a triangle. This step is crucial because failing any condition would imply the side lengths cannot create a valid triangle.

Applying the Law of Cosines

The Law of Cosines is an essential tool in solving triangles, similar to the Pythagorean theorem but applicable to all types of triangles, not just right-angled ones. It relates the length of a side of a triangle to the lengths of the other two sides and the cosine of the included angle.

In our exercise, we use it to find one of the triangle's angles, typically starting with the largest side \(c\):

• The formula is: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
• To find angle \(\angle A\), rearrange it: \[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]
• Plug in the values and solve: \[ \cos(A) = \frac{5.7^2 + 8.2^2 - 9.3^2}{2 \times 5.7 \times 8.2} = 0.1417 \]
• Convert \(\cos\) to an angle: \( A \approx \arccos(0.1417) \approx 81.84^\circ \)

The calculated angle is then used to find the remaining angles, ensuring a complete understanding of the triangle's geometry.

Techniques for Angle Calculation

When angles in a triangle are needed, especially in non-right triangles, the Law of Cosines and the angle sum property are the main techniques used. Once at least one angle is found, the rest is simpler.

After finding \(\angle A\), calculate \(\angle B\) using:

• Formula: \[ \cos(B) = \frac{a^2 + c^2 - b^2}{2ac} \]
• Values: \[ \cos(B) = \frac{9.3^2 + 8.2^2 - 5.7^2}{2 \times 9.3 \times 8.2} = 0.7963 \]
• Resulting angle: \( B \approx \arccos(0.7963) \approx 37.13^\circ \)

The last angle, \(\angle C\), takes advantage of the triangle’s angle sum property:

• Triangles have angles that add up to \(180^\circ\).
• Determine \(\angle C\): \[ C = 180^\circ - A - B = 61.03^\circ \]

These calculations cross-verify each other to ensure the triangle's validity. Understanding this holistic approach aids in solving any triangles you encounter.