Triangle Third Angle Calculator: Sides 11cm, 7cm, Angle 50°

Hey guys, let's dive into a cool geometry problem today! We've got a triangle, and we know the lengths of two of its sides – they're 11 cm and 7 cm. To make things even more interesting, we're given the included angle, which is the angle smack-dab between those two sides, and it's a neat 50°. Our mission, should we choose to accept it, is to find the third angle of this triangle. This is a classic scenario where the Law of Cosines and the Law of Sines become our best buddies. We're not looking for the third side directly, but the third angle. So, let's get our math hats on and figure this out step-by-step!

Understanding the Triangle and Our Goal

So, we're dealing with a triangle, and geometry problems like this are super common in math class and even in practical applications. We have side 'a' = 11 cm, side 'b' = 7 cm, and the angle 'C' between them is 50°. Our goal is to find angle 'A' or angle 'B' – whichever one we can tackle first. When you're given two sides and the included angle (this is often called the SAS case – Side-Angle-Side), you can uniquely determine the triangle. This means there's only one possible triangle that fits these dimensions. To find the third angle, we'll first need to find the length of the third side, let's call it 'c', using the Law of Cosines. Once we have all three sides, we can then use the Law of Sines to find one of the other angles. Alternatively, after finding the third side, we can use the Law of Cosines again to find another angle. We'll explore both paths to make sure we get this third angle nailed down.

Remember, in any triangle, the sum of all three angles is always 180°. So, if we can find two of the angles, finding the third one is a piece of cake. Our main challenge here is to use the given information effectively. The Law of Cosines is perfect for finding a side when you have SAS. It states: $c^2 = a^2 + b^2 - 2ab \cos(C)$. Once we calculate 'c', we can then use the Law of Sines: $a/\sin(A) = b/\sin(B) = c/\sin(C)$. This law is fantastic for finding angles when you know sides, or sides when you know angles. Let's get started with calculating the length of the third side, 'c', because that's our crucial first step to unlocking the third angle.

Step 1: Calculate the Third Side (Side 'c') using the Law of Cosines

Alright guys, the first major step to finding our third angle is to figure out the length of the side opposite the included angle (angle C). We'll use the Law of Cosines for this, which is a fantastic tool when you have two sides and the angle between them (SAS). The formula looks like this: $c^2 = a^2 + b^2 - 2ab \cos(C)$.

In our case, we have:

• Side $a = 11$ cm
• Side $b = 7$ cm
• Included angle $C = 50°$

Let's plug these values into the formula:

$c^2 = (11 ext{ cm})^2 + (7 ext{ cm})^2 - 2 \times (11 ext{ cm}) \times (7 ext{ cm}) \times \cos(50°)$

First, let's square the sides:

$11^2 = 121$ $7^2 = 49$

So, our equation becomes:

$c^2 = 121 + 49 - 2 \times 11 \times 7 \times \cos(50°)$

Now, let's calculate the product of $2ab$:

$2 \times 11 \times 7 = 22 \times 7 = 154$

So far, we have:

$c^2 = 121 + 49 - 154 \times \cos(50°)$

Now, we need the value of $\cos(50°)$. Using a calculator, $\cos(50°) \approx 0.6428$.

Let's substitute this value back into our equation:

$c^2 \approx 121 + 49 - 154 \times 0.6428$

$c^2 \approx 170 - 99.0092$

$c^2 \approx 70.9908$

To find the length of side 'c', we need to take the square root of $c^2$:

$c = \sqrt{70.9908}$

$c \approx 8.4256$ cm

Awesome! We've successfully calculated the length of the third side, 'c', which is approximately 8.43 cm. This is a massive step towards finding the third angle because now we have all three side lengths of the triangle. Having all the sides opens up our options for finding the angles using trigonometric laws. This side length 'c' is critical for the next phase of our calculation, where we'll use it with the Law of Sines to pinpoint one of the unknown angles. Keep this value handy, guys, it's going to be super useful!

Step 2: Find one of the other angles using the Law of Sines

Now that we've got the length of side 'c' (approximately 8.43 cm), we can use the Law of Sines to find one of the other angles. The Law of Sines is perfect for this situation because it relates the sides of a triangle to the sines of their opposite angles. The formula is: $a/\sin(A) = b/\sin(B) = c/\sin(C)$.

We know:

• Side $a = 11$ cm
• Side $b = 7$ cm
• Side $c \approx 8.43$ cm
• Angle $C = 50°$

We want to find either angle A or angle B. Let's try to find angle A. We can set up the following proportion using the Law of Sines:

$a/\sin(A) = c/\sin(C)$

Substitute the known values:

$11 ext{ cm} / \sin(A) = 8.43 ext{ cm} / \sin(50°)$

To solve for $\sin(A)$, we can rearrange the equation:

$\sin(A) = (11 ext{ cm} \times \sin(50°)) / 8.43 ext{ cm}$

First, let's find $\sin(50°)$. Using a calculator, $\sin(50°) \approx 0.7660$.

Now, plug that into the equation:

$\sin(A) = (11 \times 0.7660) / 8.43$

$\sin(A) = 8.426 / 8.43$

$\sin(A) \approx 0.9995$

To find angle A, we need to take the inverse sine (arcsin) of 0.9995:

$A = \arcsin(0.9995)$

$A \approx 88.44°$

So, angle A is approximately 88.44 degrees. This is one of our unknown angles! We're getting closer to the third angle which we are trying to find. It's super important to remember that the arcsin function can sometimes give ambiguous results, but in a triangle context, especially when dealing with side lengths derived from the Law of Cosines, we usually get the correct angle. Keep in mind, side 'a' (11 cm) is the longest side, so angle 'A' should be the largest angle, which an 88.44° angle fits perfectly with the 50° angle C.

Step 3: Calculate the Final Third Angle

Awesome work, everyone! We've successfully calculated the length of the third side ('c') and found one of the unknown angles (angle A). Now, finding the final third angle is the easiest part. Remember the fundamental rule of triangles: the sum of all interior angles in any triangle is always 180°. So, if we have angles A and C, we can easily find angle B using this relationship:

$A + B + C = 180°$

We know:

• Angle $A \approx 88.44°$
• Angle $C = 50°$

Let's plug these values in:

$88.44° + B + 50° = 180°$

Combine the known angles:

$138.44° + B = 180°$

Now, isolate angle B by subtracting 138.44° from both sides:

$B = 180° - 138.44°$

$B \approx 41.56°$

And there you have it, guys! The third angle, angle B, is approximately 41.56 degrees. So, our triangle has angles of roughly 88.44°, 41.56°, and 50°. This is the third angle we were looking for!

Let's quickly recap. We started with two sides (11 cm and 7 cm) and the included angle (50°). We used the Law of Cosines to find the third side (approx. 8.43 cm). Then, we used the Law of Sines to find one of the other angles (angle A, approx. 88.44°). Finally, using the fact that all angles add up to 180°, we calculated the last angle (angle B, approx. 41.56°). Pretty neat, right? This process shows how powerful these trigonometric laws are in solving triangle problems, especially when you need to find the third angle.

Alternative Method: Using Law of Cosines Twice

Just to show you how flexible math can be, we could also find the third angle by using the Law of Cosines twice! After finding the third side 'c' (approx. 8.43 cm) in Step 1, we can use the Law of Cosines again to find another angle, say angle A. The Law of Cosines can be rearranged to solve for an angle:

$\cos(A) = (b^2 + c^2 - a^2) / (2bc)$

Let's plug in our values:

• $a = 11$ cm
• $b = 7$ cm
• $c \approx 8.43$ cm

$\cos(A) = (7^2 + (8.43)^2 - 11^2) / (2 \times 7 \times 8.43)$

$\cos(A) = (49 + 70.9908 - 121) / (118.02)$

$\cos(A) = (119.9908 - 121) / 118.02$

$\cos(A) = -1.0092 / 118.02$

$\cos(A) \approx -0.00855$

Now, find angle A using the inverse cosine (arccos):

$A = \arccos(-0.00855)$

$A \approx 90.49°$

Wait a second! This is a different result for angle A than what we got using the Law of Sines (88.44°). Why the difference? It's likely due to rounding errors with the value of 'c' and the intermediate calculations. The Law of Sines is generally more straightforward for finding angles once you have all sides, but the Law of Cosines is more robust when dealing with obtuse angles or potential ambiguities. Let's re-evaluate the Law of Sines calculation to see if we can pinpoint the issue or if it's just rounding.

Let's re-calculate $\sin(A)$ using a more precise value for $c$: $c \approx 8.4256$

$\sin(A) = (11 \times \sin(50°)) / 8.4256$

$\sin(A) = (11 \times 0.766044) / 8.4256$

$\sin(A) = 8.426484 / 8.4256$

$\sin(A) \approx 1.00009$

Uh oh! A sine value greater than 1 is impossible. This indicates a potential issue either with the initial problem statement (though unlikely for a standard math problem) or, more likely, a subtle rounding problem in our calculation chain. Let's re-check the Law of Cosines calculation for 'c' more carefully.

$c^2 = 11^2 + 7^2 - 2(11)(7)\cos(50°)$ $c^2 = 121 + 49 - 154 \times 0.6427876$ $c^2 = 170 - 99.00929$ $c^2 = 70.99071$ $c = \sqrt{70.99071} \approx 8.425597$

Okay, 'c' is indeed around 8.4256. Let's try finding angle B using the Law of Sines:

$b/\sin(B) = c/\sin(C)$

$7 / \sin(B) = 8.4256 / \sin(50°)$

$\sin(B) = (7 \times \sin(50°)) / 8.4256$

$\sin(B) = (7 \times 0.766044) / 8.4256$

$\sin(B) = 5.362308 / 8.4256$

$\sin(B) \approx 0.63644$

$B = \arcsin(0.63644)$

$B \approx 39.53°$

Now, let's find angle A using $A + B + C = 180°$:

$A + 39.53° + 50° = 180°$

$A + 89.53° = 180°$

$A = 180° - 89.53°$

$A \approx 90.47°$

This result for angle A (90.47°) is much closer to the one we got using the Law of Cosines twice (90.49°). The slight difference is still due to cumulative rounding. This demonstrates that both methods are valid, but precision in calculations is key!

Conclusion

So, guys, we've successfully tackled a triangle problem using two different, yet complementary, trigonometric approaches! Given a triangle with two sides measuring 11 cm and 7 cm, and the included angle of 50°, we were able to find the third angle. Through careful application of the Law of Cosines to find the third side, followed by the Law of Sines (or Law of Cosines again) to find the other angles, we discovered the angles of the triangle to be approximately 90.47°, 39.53°, and the given 50°. This shows the power and versatility of trigonometry in solving geometric puzzles. Whether you're a student tackling homework or just someone who loves a good math challenge, understanding these laws is super valuable. Keep practicing, and don't be afraid to use your calculator wisely to maintain accuracy in your calculations! The key takeaway is that with SAS, you can find everything about a triangle, including that elusive third angle.