Law Of Cosines: What Can We Learn About Triangle RST?

Hey guys! Today, we're diving deep into the fascinating world of trigonometry, specifically the Law of Cosines. You know, that super handy formula that helps us find missing sides and angles in any triangle, not just the right-angled ones. We've got a specific setup here for $\triangle RST$: $5^2=7^2+3^2-2(7)(3) \cos (S)$. Our mission, should we choose to accept it, is to figure out what this equation tells us about the sides and angles of our triangle. Let's break it down and see what secrets $\triangle RST$ is hiding!

Unpacking the Law of Cosines

First off, let's refresh our memories on the general form of the Law of Cosines. For any triangle with sides $r$, $s$, and $t$, and the angle opposite side $r$ being $R$, opposite $s$ being $S$, and opposite $t$ being $T$, the law can be stated in three ways:

• $r^2 = s^2 + t^2 - 2st \cos(R)$
• $s^2 = r^2 + t^2 - 2rt \cos(S)$
• $t^2 = r^2 + s^2 - 2rs \cos(T)$

Notice the pattern here? The side squared on the left is the one opposite the angle used in the cosine term on the right. This is the golden rule, the key to unlocking the mystery!

Now, let's look at the equation given for $\triangle RST$: $5^2=7^2+3^2-2(7)(3) \cos (S)$. Compare this to our general forms. We can immediately see a strong resemblance to the second form: $s^2 = r^2 + t^2 - 2rt \cos(S)$.

Why? Because the angle on the right side of the equation is $\cos(S)$. This means the side squared on the left side of the equation must be the side opposite angle $S$. In our given equation, that side squared is $5^2$. Therefore, we can confidently say that the side opposite angle $S$, which is side $s$, must be equal to 5. So, $s=5$. This is a huge clue, guys!

Connecting the Pieces: Sides and Angles

But wait, there's more! The equation also gives us the lengths of the other two sides involved in the formula. Look at the terms $7^2$ and $3^2$. In the general form $s^2 = r^2 + t^2 - 2rt \cos(S)$, the sides $r$ and $t$ are the sides adjacent to angle $S$. In our specific equation, $5^2=7^2+3^2-2(7)(3) \cos (S)$, the numbers 7 and 3 correspond to these adjacent sides. This means the sides $r$ and $t$ of $\triangle RST$ must be 7 and 3. It doesn't matter which is which; one is $r$ and the other is $t$. So, we know that one of the sides adjacent to angle $S$ is 7, and the other is 3.

Let's recap what we've deduced so far:

• The side opposite angle $S$ is $s=5$.
• The sides adjacent to angle $S$ are $r$ and $t$, with values 7 and 3 (in any order).

Now, let's examine the options provided to see which one aligns with our findings. We are looking for a scenario where side $s$ is 5, and the other two sides ($r$ and $t$) are 7 and 3.

• A. $r=5$ and $t=7$: If $r=5$ and $t=7$, then side $s$ (opposite angle $S$) would be calculated using $s^2 = 5^2 + 7^2 - 2(5)(7) \cos(S)$. This doesn't match our given equation where the side opposite $S$ is 5.
• B. $r=3$ and $t=3$: If $r=3$ and $t=3$, then side $s$ would be $s^2 = 3^2 + 3^2 - 2(3)(3) \cos(S)$. Again, this doesn't align with our side $s$ being 5.
• C. $s=7$ and $t=5$: If $s=7$ and $t=5$, the Law of Cosines for angle $S$ would be $s^2 = r^2 + t^2 - 2rt \cos(S)$, so $7^2 = r^2 + 5^2 - 2(r)(5) \cos(S)$. This doesn't fit our initial setup.
• D. $s=5$ and $t=3$: This option states that side $s=5$ and side $t=3$. If $s=5$ and $t=3$, then the Law of Cosines involving angle $S$ would be $s^2 = r^2 + t^2 - 2rt \cos(S)$. Substituting our known values, we get $5^2 = r^2 + 3^2 - 2(r)(3) \cos(S)$. For this to match the given equation $5^2=7^2+3^2-2(7)(3) \cos (S)$, we need $r^2=7^2$ and $r=7$. This fits perfectly! So, if $s=5$ and $t=3$, then $r$ must be 7. This matches our deduction that the sides adjacent to angle $S$ are 7 and 3, and the side opposite angle $S$ is 5.

Therefore, the statement that could be true about $\triangle RST$ is that $s=5$ and $t=3$ (which implies $r=7$).

The Importance of Context in Trigonometry

The Law of Cosines is an incredibly powerful tool in trigonometry, guys, and understanding how to read and interpret its form is crucial. The setup $5^2=7^2+3^2-2(7)(3) \cos (S)$ isn't just a random string of numbers; it's a direct representation of a specific triangle's properties. The key takeaway is always to link the side squared on the left to the angle within the cosine term on the right. In this case, $5^2$ is on the left, and $\cos(S)$ is on the right. This unequivocally tells us that the side opposite angle $S$, denoted as $s$, must be 5. This is the foundational piece of information we extract.

Following this, we look at the other two terms in the equation: $7^2$ and $3^2$. These represent the squares of the lengths of the other two sides of the triangle, $r$ and $t$. The term $-2(7)(3) \cos(S)$ confirms that these sides, $r$ and $t$, are indeed the ones adjacent to angle $S$. So, we know that the set of side lengths for $\triangle RST$ includes 5 (opposite angle $S$) and that the other two sides are 7 and 3. The beauty of the Law of Cosines is its consistency. It applies to all triangles, and its structure provides direct clues about the relationships between sides and angles.

When presented with multiple-choice options, the process becomes one of elimination and verification. We've already established that $s=5$. This immediately helps us rule out options A, B, and C, as they either assign a different value to $s$ or don't align with the side lengths we've deduced. Option D, $s=5$ and $t=3$, aligns perfectly with our findings. If $s=5$ and $t=3$, then the Law of Cosines for angle $S$ is $s^2 = r^2 + t^2 - 2rt \cos(S)$. Plugging in $s=5$ and $t=3$, we get $5^2 = r^2 + 3^2 - 2(r)(3) \cos(S)$. For this to match the given equation $5^2=7^2+3^2-2(7)(3) \cos (S)$, it necessitates that $r^2=7^2$ and $r=7$. This confirms that the sides adjacent to angle $S$ are indeed 7 and 3, which is exactly what the original equation implies. So, option D is the correct deduction about $\triangle RST$. It highlights how a single trigonometric equation can encapsulate a wealth of information about a geometric figure, allowing us to deduce its properties with precision. Remember, guys, always trust the formula and break it down piece by piece!

Conclusion: Decoding $\triangle RST$

So, there you have it, math enthusiasts! By carefully dissecting the given Law of Cosines equation, $5^2=7^2+3^2-2(7)(3) \cos (S)$, we've successfully decoded the characteristics of $\triangle RST$. The equation structure is our roadmap. The side squared on the left ($5^2$) directly corresponds to the side opposite the angle in the cosine term ($\cos(S)$), meaning side $s$ must be 5. The other two numbers in the equation (7 and 3) represent the lengths of the sides adjacent to angle $S$, which are sides $r$ and $t$. Therefore, we know that $s=5$, and the pair of sides $(r, t)$ is $(7, 3)$ or $(3, 7)$.

When we evaluate the given options, option D, $s=5$ and $t=3$, stands out as the only one consistent with our deductions. If $s=5$ and $t=3$, then the Law of Cosines applied to angle $S$ is $s^2 = r^2 + t^2 - 2rt \cos(S)$. Substituting the values, we get $5^2 = r^2 + 3^2 - 2(r)(3) \cos(S)$. Comparing this to the original equation $5^2=7^2+3^2-2(7)(3) \cos (S)$, it becomes clear that $r^2$ must equal $7^2$, and thus $r=7$. This confirms that the sides adjacent to angle $S$ are indeed 7 and 3, and the side opposite is 5. This comprehensive understanding allows us to confidently select the correct answer and appreciate the elegance of trigonometric relationships in describing geometric shapes. Keep practicing, keep questioning, and you'll master these concepts in no time!