Problem 7 - Basic Engineering Mathematics

Question

Express (4.5, 5.16 rad) in Cartesian co-ordinates A sketch showing the position \((4.5,5.16 \mathrm{rad}\) ) is shown in Fig. 21.9. $$ x=r \cos \theta=4.5 \cos 5.16=1.948 $$ which corresponds to length \(O A\) in Fig. \(21.9\).

Step-by-Step Solution

Answer

The Cartesian coordinates are approximately (1.948, -4.063).

Step 1: Understand the Polar Coordinates

In this exercise, we are given the polar coordinates \((4.5, 5.16)\). This means the point is at a distance \(r = 4.5\) from the origin and makes an angle \(\theta = 5.16\) radians with the positive x-axis.

Step 2: Convert to Cartesian Coordinates

We use the formulas for conversion: \[ x = r \cos \theta \]\[ y = r \sin \theta \]

Step 3: Calculate x-coordinate

Using the formula for \(x\):\[ x = 4.5 \cos 5.16 \approx 1.948 \]Here, \(\cos 5.16\) was used to find the x-coordinate, confirming the calculation \(x = 1.948\).

Step 4: Calculate y-coordinate

Now calculate the y-coordinate using the formula:\[ y = 4.5 \sin 5.16 \]Solving this will give the y-coordinate value, which you must compute using a calculator.

Step 5: Final Cartesian Coordinates

Combine the x and y coordinates to find the Cartesian coordinates. If you computed \( y \) to be approximately \(-4.063\), then the Cartesian coordinates are \((1.948, -4.063)\).

Key Concepts

Polar Coordinates

Polar coordinates provide a different way to locate points in a plane. Unlike Cartesian coordinates, which use horizontal (x) and vertical (y) values, polar coordinates use a distance and an angle. Here, each point is described by:

• The radius ( "): the distance from the origin.
• The angle ( heta"): the direction from the positive x-axis.

Polar coordinates are practical for problems involving circular motion or rotation. They can simplify calculations by directly representing periodic motion, which is common in physics and engineering. Understanding polar coordinates is key to visualizing problems with symmetry or circular paths, and they offer an intuitive approach in some scenarios.

Coordinate Conversion

The process of coordinate conversion involves changing a point from polar coordinates to Cartesian coordinates, or vice versa. This is essential because different problems are easier to solve in different coordinate systems. In our example, we convert from polar (4.5, 5.16) to Cartesian coordinates.
The conversion formulae are:

• To find the x-coordinate: \[ x = r \cos \theta \]
• To find the y-coordinate: \[ y = r \sin \theta \]

These transformations leverage trigonometry to translate circular motion into linear measures, bridging the gap between the two systems effectively.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in converting between polar and Cartesian coordinates. They help translate an angular position into linear dimensions.
For example, cosine helps determine the x-component of a point based on the angle. In this case, the formula \( x = r \cos \theta \) provides the horizontal distance.

• Sine (\(\sin\theta\)): calculates the y-coordinate, representing vertical projection.
• Cosine (\(\cos\theta\)): computes the x-coordinate, indicating horizontal displacement.

Understanding basic properties of sine and cosine — like how they oscillate between -1 and 1 — can also benefit a broader range of math problem solving and applications.

Math Problem Solving

Solving math problems involving coordinate conversion often requires breaking down the task into clear, manageable steps. First, accurately interpret the given data, whether in polar or Cartesian form. Then, apply direct conversion formulae while ensuring precision with each computation.
Tools such as a calculator can support finding precise trigonometric values; however, it's also essential to thoroughly sketch and visualize graphs or diagrams when feasible. This comprehensive approach facilitates deeper insight into the problem, highlighting connections between geometric figures and algebraic expressions, enhancing both understanding and accuracy.