Problem 55 - Trigonometry

Question

Assume vector \(\mathbf{V}\) is in standard position, has the given magnitude, and that \(\theta\) is the angle \(\mathbf{V}\) makes with the positive \(x\)-axis. Write \(\mathbf{V}\) in vector component form \(a \mathbf{i}+b \mathbf{j}\), and approximate your values to two significant digits. $$|\mathbf{V}|=5.8, \theta=71^{\circ}$$

Step-by-Step Solution

Answer

\(\mathbf{V} \approx 1.9 \mathbf{i} + 5.5 \mathbf{j}\).

Step 1: Understand the Problem

We need to find the component form of vector \(\mathbf{V}\) given its magnitude \(|\mathbf{V}|=5.8\) and the angle \(\theta=71^{\circ}\) with the positive \(x\)-axis. The vector component form is \(a \mathbf{i} + b \mathbf{j}\).

Step 2: Use Trigonometric Functions to Find Components

Recall that if a vector \(\mathbf{V}\) has a magnitude \(r\) and makes an angle \(\theta\) with the positive \(x\)-axis, its components are given by: \(a = r \cos \theta\) and \(b = r \sin \theta\).

Step 3: Calculate the \(x\)-Component

Substitute the known values into the cosine formula: \(a = 5.8 \cos 71^{\circ}\). Calculate this value: \(a = 5.8 \times 0.3256 \approx 1.889\). Rounding to two significant digits, \(a \approx 1.9\).

Step 4: Calculate the \(y\)-Component

Substitute the known values into the sine formula: \(b = 5.8 \sin 71^{\circ}\). Calculate this value: \(b = 5.8 \times 0.9455 \approx 5.483\). Rounding to two significant digits, \(b \approx 5.5\).

Step 5: Write the Vector in Component Form

Combine the results to write the vector \(\mathbf{V}\) in component form. Using the calculated values, we have: \( \mathbf{V} \approx 1.9 \mathbf{i} + 5.5 \mathbf{j}\).

Key Concepts

Understanding Trigonometric Functions in Vectors

Trigonometric functions are essential tools in the world of vectors, helping us determine components along the coordinate axes. When a vector is positioned at an angle with respect to the positive x-axis, the trigonometric functions sine and cosine come into play. The cosine of the angle helps to find the vector's horizontal component, representing the influence along the x-axis.

On the other hand, the sine of the angle gives us the vertical component or the effect along the y-axis. In our specific exercise, with a magnitude of 5.8 and an angle of 71 degrees, we use these functions as follows:

• Cosine for the x-component: \( a = r \cos \theta \)
• Sine for the y-component: \( b = r \sin \theta \)

When you input these values into a calculator, ensure your calculator is set to degrees rather than radians to avoid any errors. By harnessing these trigonometric functions, you can accurately dissect the vector's direction into its components.

Magnitude and Direction of Vectors

Understanding the magnitude and direction of vectors is fundamental to mastering vector mathematics. The magnitude, or length, of a vector determines how much force or motion it represents. In our exercise, the magnitude 5.8 indicates how strong or significant the vector is in real-world applications.

The direction, designated by the angle \(\theta\), dictates the vector's orientation in the coordinate plane. Here, an angle of 71 degrees indicates the direction counterclockwise from the positive x-axis, providing a sense of where the vector "points."

To visualize this, imagine standing on the x-axis and turning 71 degrees toward the y-axis—that's the vector's trajectory. By splitting the vector into its x and y components using the magnitude and direction, we can better understand its impact in each direction:

• Magnitude gives us the vector's overall length
• Direction provides the angle of orientation in the plane

This separation is crucial for tasks such as navigation, physics calculations, and computer graphics, where knowing the vector's precise contribution on each axis is key.

Approximating Values with Two Significant Digits

In mathematics, approximation helps make complex calculations more manageable and results easier to interpret. Approximating values, like rounding them to two significant digits, is particularly helpful in simplifying the results while retaining significant digits that reflect precision.

In our exercise, after calculating the exact components using trigonometric values, results such as 1.889 or 5.483 are rounded to 1.9 and 5.5, respectively. This practice enhances clarity, especially in situations where further accuracy isn't critical to interpretation or application.

Here's how you might decide on approximating your results:

• Determine if precision beyond two digits is necessary for your problem context.
• If not, round your result up or down to the nearest value that maintains significant meaning.

This focus on abbreviated precision helps keep calculations simple and concise, which is especially useful when presenting or using the results in practical applications. Approximating enables a balance between comprehensibility and detail, ensuring that the key characteristics remain highlighted.