Solving Tan(34°) = 9/u: A Step-by-Step Guide

Are you wrestling with the trigonometric equation tan(34°) = 9/u and need a clear, step-by-step solution? You've come to the right place! This comprehensive guide will walk you through the process of solving for u, ensuring you understand each step along the way. We'll break down the equation, explain the trigonometric concepts involved, and provide the final answer rounded to one decimal place. Understanding trigonometry can be tricky, but with a clear explanation and a bit of practice, you'll be solving these problems in no time. Let's dive in and conquer this mathematical challenge together!

Understanding the Problem: tan(34°) = 9/u

Before we jump into the solution, let's make sure we understand the problem. The equation we're tackling is tan(34°) = 9/u. This involves a trigonometric function, tangent (tan), and an unknown variable, u, that we need to find. Understanding the interplay between trigonometric functions and algebraic manipulation is crucial for solving this problem. Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The tangent function, in particular, relates the opposite and adjacent sides of a right-angled triangle to a specific angle. In this equation, tan(34°) represents the tangent of an angle of 34 degrees. This value is a ratio, and we can find it using a calculator. The right side of the equation, 9/u, represents a fraction where 9 is the numerator and u is the denominator. Our goal is to isolate u on one side of the equation to find its value. This involves using algebraic principles to manipulate the equation until u stands alone. Remember, the key to solving any equation is to maintain balance. Whatever operation you perform on one side, you must also perform on the other side to keep the equation true. This principle will guide us as we work through the steps to find the value of u. So, let's break down the process and solve for u step by step!

Step 1: Isolating u

Our first goal in solving tan(34°) = 9/u is to isolate the variable u. Currently, u is in the denominator of the fraction on the right side of the equation, which makes it difficult to work with directly. To get u out of the denominator, we need to perform an algebraic manipulation that moves it to the numerator. The most straightforward way to do this is to multiply both sides of the equation by u. This is a valid operation because, as we discussed earlier, performing the same operation on both sides of an equation maintains the equality. Multiplying both sides of tan(34°) = 9/u by u gives us: u * tan(34°) = u * (9/u) On the right side, the u in the numerator and the u in the denominator cancel each other out, simplifying the equation. This leaves us with: u * tan(34°) = 9 Now, u is multiplied by tan(34°), which is a numerical value (we can calculate it using a calculator). To isolate u completely, we need to undo this multiplication. The inverse operation of multiplication is division, so our next step will involve dividing both sides of the equation by tan(34°). This will finally leave u alone on one side, allowing us to determine its value. Remember, the key is to systematically apply algebraic operations to both sides, gradually simplifying the equation until the variable we're interested in is isolated. Now that we have u * tan(34°) = 9, let's move on to the next step and divide both sides by tan(34°) to solve for u.

Step 2: Dividing by tan(34°)

Having successfully moved u to the numerator, we now have the equation u * tan(34°) = 9. Our next step is to isolate u completely by undoing the multiplication by tan(34°). As we discussed, the inverse operation of multiplication is division, so we need to divide both sides of the equation by tan(34°). This will effectively cancel out tan(34°) on the left side, leaving u by itself. Performing this division, we get: (u * tan(34°)) / tan(34°) = 9 / tan(34°) On the left side, tan(34°) in the numerator and the denominator cancel each other out, simplifying the equation to: u = 9 / tan(34°) Now we have u expressed as a fraction where the numerator is 9 and the denominator is tan(34°). To find the numerical value of u, we need to calculate tan(34°) and then perform the division. This is where a calculator comes in handy. Make sure your calculator is in degree mode, as we are dealing with an angle measured in degrees. Once you have the value of tan(34°), you can divide 9 by that value to find u. Remember, we need to round our final answer to one decimal place, as specified in the problem. So, after performing the division, we'll look at the second decimal place to determine whether to round up or down. This step of dividing by tan(34°) is crucial in isolating u and bringing us closer to the solution. Let's move on to the next step, where we'll calculate tan(34°) and find the value of u.

Step 3: Calculating tan(34°) and Finding u

With the equation u = 9 / tan(34°), we're now ready to calculate the value of u. The first part of this step involves finding the value of tan(34°). This is where we'll use a calculator. Ensure your calculator is set to degree mode (usually indicated by a