Finding Side Length Using Cosine: Step-by-Step Solution

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Finding Side Length Using Cosine: A Step-by-Step Solution

Hey guys! Let's break down this problem where we need to find the length of a side using the cosine function. We're given the equation cos(35°) = a/25, and our mission is to figure out the length of side BC (which we're calling 'a' in this case). Don't worry, it's easier than it looks! We'll go through it step-by-step, so you can totally nail it.

Understanding the Cosine Function

Before we dive into solving the equation, let's quickly recap what cosine actually means in trigonometry. Remember SOH CAH TOA? This handy acronym helps us remember the relationships between angles and sides in a right triangle.

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

In our problem, we're dealing with cosine (CAH). So, cos(35°) represents the ratio of the adjacent side to the hypotenuse in a right triangle where one of the angles is 35 degrees. The adjacent side is the side next to the angle (not the hypotenuse), and the hypotenuse is the longest side, opposite the right angle. In our equation, 'a' represents the length of the adjacent side (BC), and 25 represents the length of the hypotenuse.

Solving for the Unknown Side

Now that we understand the cosine function, let's solve for 'a'. Our equation is cos(35°) = a/25. To isolate 'a', we need to get rid of the fraction. We can do this by multiplying both sides of the equation by 25. This is a fundamental rule of algebra: whatever you do to one side of the equation, you have to do to the other side to keep it balanced.

So, multiplying both sides by 25, we get:

25 * cos(35°) = a

Now we're getting somewhere! We've got 'a' all by itself on one side of the equation. The next step is to actually calculate the value of 25 * cos(35°). This is where your calculator comes in handy. Make sure your calculator is in degree mode (not radians) because our angle is given in degrees. If it's in radian mode, you'll get the wrong answer. Usually, there's a setting you can change, often labeled "DEG" or "RAD".

Using a Calculator to Find the Cosine

Using your calculator, find the cosine of 35 degrees. You should get a decimal value close to 0.819. This is the ratio of the adjacent side to the hypotenuse for a 35-degree angle. Now, multiply this value by 25:

25 * 0.819 ≈ 20.475

So, we've found that 'a' is approximately equal to 20.475. But wait, the problem asks us to round to the nearest tenth. That means we need to look at the digit in the hundredths place (the second digit after the decimal point) to decide whether to round up or down. In this case, the hundredths digit is 7, which is greater than or equal to 5. So, we round up the tenths digit.

Rounding to the Nearest Tenth

Rounding 20.475 to the nearest tenth gives us 20.5. Therefore, the length of side BC (represented by 'a') is approximately 20.5 units. Remember to include the units if they are given in the problem! In this case it's in.

So, the final answer is 20.5 in. Option B is the correct answer.

Why the Other Options Are Incorrect

It's always helpful to understand why the other answer choices are wrong. This helps solidify your understanding of the concept and prevent similar mistakes in the future. Let's briefly look at why the other options might be incorrect:

  • A. 14.3 in: This answer might result from using the sine function instead of the cosine function (sin(35°) = opposite / hypotenuse). If you mistakenly calculated 25 * sin(35°), you'd get a value close to 14.3.
  • C. 21.3 in: This answer might result from an error in the calculator calculation or a misunderstanding of which side is adjacent to the angle.
  • D. 22.6 in: This answer is less likely but could stem from a combination of errors, such as using an incorrect trigonometric function or making a mistake during the multiplication or rounding process.

Key Takeaways

  • Understand SOH CAH TOA: This is fundamental for solving trigonometry problems.
  • Identify the correct trigonometric function: In this case, we used cosine because we were dealing with the adjacent side and the hypotenuse.
  • Set up the equation correctly: Make sure you place the values in the correct positions in the equation.
  • Use your calculator carefully: Ensure it's in degree mode and double-check your calculations.
  • Round appropriately: Pay attention to the instructions and round to the specified decimal place.

Practice Makes Perfect

Trigonometry can seem tricky at first, but the more you practice, the easier it becomes. Try working through similar problems, changing the angle or the side you need to find. You can also use online resources or textbooks for additional practice problems.

Keep practicing, and you'll become a trigonometry pro in no time! Remember, understanding the concepts is key, so don't just memorize formulas – think about what each function represents and how it relates to the sides of a right triangle.

I hope this explanation helped you guys understand how to solve this problem. If you have any questions, feel free to ask! And remember, math can be fun if you approach it step by step and break it down into smaller, manageable parts.