Calculating Angles In Triangle ABC A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of triangles, specifically triangle ABC, where we're given the side lengths a = 6.5 cm, b = 8.5 cm, and c = 4.5 cm. Our mission? To calculate all the angles of this triangle. We'll be tackling this step-by-step, ensuring you understand not just the how, but also the why behind each calculation. So, buckle up and let's get started!

i. Finding the Largest Angle: A Deep Dive

When it comes to finding the largest angle in a triangle, the key concept to remember is that the largest angle is always opposite the longest side. In our triangle ABC, side b (8.5 cm) is the longest side. Therefore, angle B is the largest angle. To calculate angle B, we'll employ the Law of Cosines, a fundamental tool in trigonometry that relates the sides and angles of any triangle. The Law of Cosines states:

$$b^2 = a^2 + c^2 - 2ac \cdot cos(B)$$

Let's break down this formula and see how it applies to our problem. The formula essentially tells us that the square of one side of a triangle (b in this case) is equal to the sum of the squares of the other two sides (a and c) minus twice the product of those sides and the cosine of the angle opposite the first side (angle B). Now, let's plug in the values we know:

$$(8.5)^2 = (6.5)^2 + (4.5)^2 - 2 \cdot (6.5) \cdot (4.5) \cdot cos(B)$$

This equation has only one unknown, cos(B), which we can solve for. First, let's simplify the equation:

$$72.25 = 42.25 + 20.25 - 58.5 \cdot cos(B)$$

Combine the constants:

$$72.25 = 62.5 - 58.5 \cdot cos(B)$$

Now, isolate the term with cos(B):

$$9.75 = -58.5 \cdot cos(B)$$

Divide both sides by -58.5 to solve for cos(B):

$$cos(B) = -0.16666666666$$

To find the angle B, we need to take the inverse cosine (also known as arccos or cos⁻¹) of -0.16666666666. Make sure your calculator is in degree mode for this calculation! Using a calculator, we find:

$$B = arccos(-0.16666666666) \approx 99.6 \text{ degrees}$$

Therefore, the largest angle in triangle ABC, angle B, is approximately 99.6 degrees. This makes sense intuitively – since side b is significantly longer than sides a and c, we expect the opposite angle to be quite large, even obtuse (greater than 90 degrees).

Understanding the Law of Cosines is crucial here. It's not just about plugging in numbers; it's about understanding the relationship between side lengths and angles in a triangle. The Law of Cosines is a powerful tool that can be applied to any triangle, regardless of whether it's a right triangle or not. Remember, the largest angle is always opposite the longest side, a principle that helps you quickly identify which angle to focus on first.

ii. Calculating the Smallest Angle: A Similar Approach

Now that we've conquered the largest angle, let's move on to calculating the smallest angle. Following the same logic as before, the smallest angle is opposite the shortest side. In triangle ABC, side c (4.5 cm) is the shortest side, so angle C is the smallest angle. We'll once again use the Law of Cosines, but this time, we'll rearrange it to solve for angle C:

$$c^2 = a^2 + b^2 - 2ab \cdot cos(C)$$

Plugging in the values:

$$(4.5)^2 = (6.5)^2 + (8.5)^2 - 2 \cdot (6.5) \cdot (8.5) \cdot cos(C)$$

Simplify the equation:

$$20.25 = 42.25 + 72.25 - 110.5 \cdot cos(C)$$

Combine the constants:

$$20.25 = 114.5 - 110.5 \cdot cos(C)$$

Isolate the term with cos(C):

$$-94.25 = -110.5 \cdot cos(C)$$

Divide both sides by -110.5 to solve for cos(C):

$$cos(C) = 0.85294117647$$

To find angle C, take the inverse cosine:

$$C = arccos(0.85294117647) \approx 31.45 \text{ degrees}$$

Therefore, the smallest angle in triangle ABC, angle C, is approximately 31.45 degrees. This angle is acute (less than 90 degrees), which makes sense given that side c is the shortest side. The calculation process is very similar to finding the largest angle, emphasizing the versatility of the Law of Cosines.

An alternative approach to finding the smallest angle, once you've found the largest, is to use the Law of Sines. However, the Law of Cosines is generally preferred when you have all three sides because it avoids the ambiguity that can arise with the Law of Sines (where there might be two possible angles for a given sine value). Using the Law of Cosines directly ensures you get the correct angle without needing to consider multiple possibilities. Remember, choosing the right tool for the job is key in trigonometry!

iii. Determining the Third Angle: The Angle Sum Property

Finally, let's calculate the third angle. Now that we know angle B (99.6 degrees) and angle C (31.45 degrees), we can find angle A using a fundamental property of triangles: the sum of the angles in any triangle is always 180 degrees. This is known as the Angle Sum Property of Triangles.

So, we have:

$$A + B + C = 180 \text{ degrees}$$

Plug in the values we know:

$$A + 99.6 + 31.45 = 180$$

Combine the constants:

$$A + 131.05 = 180$$

Subtract 131.05 from both sides to solve for A:

$$A = 180 - 131.05 \approx 48.95 \text{ degrees}$$

Therefore, the third angle, angle A, is approximately 48.95 degrees. This angle is also acute, fitting our expectation given the relative lengths of sides a and b. The Angle Sum Property is a simple yet powerful tool that allows us to find the third angle of a triangle once we know the other two.

This final step highlights the interconnectedness of angles in a triangle. Knowing two angles automatically determines the third, a crucial concept for solving various triangle-related problems. It's also a good practice to check if your calculated angles add up to 180 degrees, providing a quick way to verify your work and catch any potential errors.

In conclusion, we've successfully calculated all the angles of triangle ABC using the Law of Cosines and the Angle Sum Property. By understanding these principles and applying them systematically, you can confidently tackle a wide range of triangle problems. Keep practicing, and you'll become a master of triangle trigonometry in no time! Remember guys, understanding the why behind the formulas is just as important as knowing the how. Good luck, and happy calculating!