Finding Sin(BAC) In Triangle ABC: A Step-by-Step Guide

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Hey guys! Today, we're diving into a cool geometry problem that involves finding the sine of an angle within a triangle. Specifically, we're going to figure out how to calculate sin(BAC) in triangle ABC, given some side lengths and the area of the triangle. It might sound a bit tricky at first, but trust me, we'll break it down step by step so it's super easy to understand.

Understanding the Problem

Before we jump into the solution, let's make sure we fully grasp what the problem is asking. We have a triangle, helpfully named ABC. We know the lengths of two sides, AB and AC, which are 12 cm and 15 cm respectively. We also know that the area enclosed by this triangle is 54 square centimeters. Our mission, should we choose to accept it (and we do!), is to find the value of sin(BAC). This essentially means we need to determine the sine of the angle formed at vertex A, nestled between sides AB and AC.

The sine of an angle, if you remember your trigonometry, is a ratio that relates the angle to the sides of a right-angled triangle. But don't worry if we don't have a right-angled triangle here. We have a formula that will save the day, one that cleverly connects the area of any triangle to the sine of its angles and the lengths of its sides. This is where the magic happens, guys!

To really nail this, it's always a good idea to visualize the triangle. Imagine triangle ABC in your mind, or even better, sketch it out on a piece of paper. Label the sides AB as 12 cm and AC as 15 cm. Think about the angle BAC sitting there, waiting for us to discover its sine. And remember, that area of 54 cm² is a crucial piece of information – it's the bridge that links the side lengths to the sine of the angle. So, with our problem clearly in sight, let's arm ourselves with the right formula and start cracking this geometric puzzle!

The Area Formula to the Rescue

The key to solving this problem lies in a neat little formula that connects the area of a triangle to the lengths of two of its sides and the sine of the included angle. This formula is a powerhouse in trigonometry and geometry, and it's exactly what we need to find sin(BAC). Here's how it looks:

Area = (1/2) * a * b * sin(C)

Where:

  • Area is the area of the triangle.
  • a and b are the lengths of two sides of the triangle.
  • C is the angle included between sides a and b.

Now, let's translate this into our specific problem. In triangle ABC, we know the lengths of sides AB and AC, and we want to find sin(BAC). So, we can rewrite the formula like this:

Area(ABC) = (1/2) * |AB| * |AC| * sin(BAC)

See how smoothly that fits? We've simply replaced the generic terms with the specifics of our triangle. This formula is our golden ticket here. We already know the area of triangle ABC (54 cm²), the length of AB (12 cm), and the length of AC (15 cm). The only missing piece of the puzzle is sin(BAC), which is exactly what we're trying to find!

Think of this formula as a perfect recipe. We have all the ingredients except one, and we know the final dish we want to create (the area). By plugging in the ingredients we have, we can use the formula to back-calculate the missing ingredient (sin(BAC)). It's like a mathematical treasure hunt, and the formula is our map. So, now that we have the map, let's plug in the values and get closer to finding that treasure!

Plugging in the Values

Alright, guys, this is where the fun really begins! We're going to take the formula we discussed earlier and plug in the values we know. This is like fitting the pieces of a jigsaw puzzle together – once we have all the numbers in the right spots, the solution will start to reveal itself.

We have our formula:

Area(ABC) = (1/2) * |AB| * |AC| * sin(BAC)

And we know:

  • Area(ABC) = 54 cm²
  • |AB| = 12 cm
  • |AC| = 15 cm

Let's substitute these values into the formula:

54 cm² = (1/2) * 12 cm * 15 cm * sin(BAC)

Now we have a simple equation with one unknown, sin(BAC). The next step is to simplify this equation and isolate sin(BAC) so we can find its value. Think of it like untangling a knot – we need to carefully work through the equation, step by step, until we have sin(BAC) all by itself on one side. We're essentially using our mathematical superpowers to solve for the unknown! So, let's put on our math hats and get to work simplifying this equation. We're on the home stretch now!

Simplifying the Equation

Okay, let's roll up our sleeves and get to simplifying this equation! We've got:

54 = (1/2) * 12 * 15 * sin(BAC)

First, let's multiply the numbers on the right side. We have (1/2) * 12 * 15. We can easily compute this by first multiplying 12 and 15, which gives us 180. Then, we multiply that by (1/2), which is the same as dividing by 2. So, 180 / 2 = 90. This simplifies our equation to:

54 = 90 * sin(BAC)

Now, we want to isolate sin(BAC). To do this, we need to get rid of the 90 that's being multiplied by sin(BAC). The opposite of multiplication is division, so we'll divide both sides of the equation by 90. This keeps the equation balanced and helps us get closer to our answer:

54 / 90 = sin(BAC)

We can simplify the fraction 54/90 by finding the greatest common divisor (GCD) of 54 and 90. The GCD is 18. Dividing both the numerator and denominator by 18, we get:

(54 / 18) / (90 / 18) = 3 / 5

So, our equation now looks like this:

sin(BAC) = 3/5

Boom! We've done it! We've successfully isolated sin(BAC) and found its value. It's like cracking a secret code, and the answer is a beautifully simple fraction: 3/5. But let's not stop here – we'll confirm our answer and make sure we understand what it means in the context of the problem.

The Final Answer

Alright, guys, after all that mathematical maneuvering, we've arrived at our final answer! We found that:

sin(BAC) = 3/5

This means that the sine of the angle BAC in triangle ABC is 3/5. We successfully used the area formula and our algebra skills to solve for the unknown. It's always a great feeling when a problem clicks into place like this, isn't it?

But before we celebrate too much, let's just take a moment to recap what we did. We started with a triangle where we knew two side lengths and the area. We wanted to find the sine of the angle between those two sides. We dusted off our trusty area formula, plugged in the known values, simplified the equation, and voilà, we had our answer!

And that's how you tackle a problem like this! Remember, geometry and trigonometry can seem intimidating at first, but by breaking things down into smaller steps and using the right formulas, you can conquer any challenge. Keep practicing, keep exploring, and most importantly, keep having fun with math!

So, to definitively answer the question: The value of sin(BAC) in triangle ABC, where |AB| = 12 cm, |AC| = 15 cm, and the area of triangle ABC is 54 cm², is 3/5. You nailed it!