Proving the Angles of a Triangle with Sides 3, 7, and 8 Form an Arithmetic Sequence

The challenge presented here is to show that the angles of a triangle, with sides of 3, 7, and 8 units respectively, form an arithmetic sequence. This involves understanding the relationship between the sides and angles of the triangle, particularly through the application of the Law of Cosines. Let's break this down into clear and understandable steps.

Understanding the Triangle

The sides and their corresponding angles in this triangle are as follows:

Side a 3 (opposite angle A) Side b 7 (opposite angle B) Side c 8 (opposite angle C)

Using the Law of Cosines

The Law of Cosines is a powerful tool for solving triangles and can be used to find the angles when the sides are known. The formula is:

c^2 a^2 b^2 - 2ab cos(C)

We will use this formula to find the angles of the triangle step-by-step.

Step 1: Finding Angle C

First, we apply the Law of Cosines to find angle C:

cos(C) frac{a^2 b^2 - c^2}{2ab} frac{3^2 7^2 - 8^2}{2 cdot 3 cdot 7} frac{9 49 - 64}{42} frac{-6}{42} -frac{1}{7}

Now, we find the angle C:

C cos^{-1}left(-frac{1}{7}right)

Step 2: Finding Angle A

Next, we use the Law of Cosines to find angle A:

a^2 b^2 c^2 - 2bc cos(A)

Rearranging to solve for (cos(A)):

cos(A) frac{b^2 c^2 - a^2}{2bc} frac{7^2 8^2 - 3^2}{2 cdot 7 cdot 8} frac{49 64 - 9}{112} frac{104}{112} frac{13}{14}

Now, we find the angle A:

A cos^{-1}left(frac{13}{14}right)

Step 3: Finding Angle B

Finally, we use the Law of Cosines to find angle B:

b^2 a^2 c^2 - 2ac cos(B)

Rearranging to solve for (cos(B)):

cos(B) frac{a^2 c^2 - b^2}{2ac} frac{3^2 8^2 - 7^2}{2 cdot 3 cdot 8} frac{9 64 - 49}{48} frac{24}{48} frac{1}{2}

Now, we find the angle B:

B cos^{-1}left(frac{1}{2}right) 60^circ

Verifying the Angles Form an Arithmetic Sequence

To check if the angles form an arithmetic sequence, we calculate the angles in degrees:

A cos^{-1}left(frac{13}{14}right) (approximate value needed) B 60 C 180 - A - 60

We can verify if the angles form an arithmetic sequence by checking if: 2B A C

The final step is to calculate the value of A and C, then check the arithmetic progression condition. If the condition is satisfied, it confirms that the angles indeed form an arithmetic sequence.

This problem demonstrates the power of the Law of Cosines in solving complex geometric problems. Understanding and applying these mathematical principles accurately can provide insights into the relationships between the sides and angles of triangles.

References:

Wikipedia: Law of Cosines Math is Fun: Geometry and Trigonometry Khan Academy: Law of Cosines