Problem 4 - Precalculus Mathematics for Calculus

Question

Determine the \(X Y\)-coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$(-2,1), \quad \phi=30^{\circ}$$

Step-by-Step Solution

Answer

The new coordinates are approximately (-1.232, 1.866).

Step 1: Understanding the Problem

You are given a point \((-2,1)\) and asked to find its new coordinates after a rotation of \(30^{\circ}\) about the origin. The coordinate axes are rotated, not the point itself.

Step 2: Understanding Rotation Transformation Formulas

To find the new coordinates after rotation, we use the formulas:\[x' = x \cos\phi + y \sin\phi\]\[y' = -x \sin\phi + y \cos\phi\]where \( (x, y) \) are the original coordinates and \( (x', y') \) are the new coordinates.

Step 3: Applying Rotation Formulas

Substitute the given values into the formulas to calculate the rotated coordinates:\[x' = -2 \cdot \cos(30^{\circ}) + 1 \cdot \sin(30^{\circ})\]\[y' = -(-2) \cdot \sin(30^{\circ}) + 1 \cdot \cos(30^{\circ})\]

Step 4: Calculating Trigonometric Values

From trigonometry, we know:\[\cos(30^{\circ}) = \frac{\sqrt{3}}{2}, \quad \sin(30^{\circ}) = \frac{1}{2}\]

Step 5: Substituting Trigonometric Values into the Equations

Substitute \(\cos(30^{\circ})\) and \(\sin(30^{\circ})\) into the formulas:\[x' = -2 \cdot \frac{\sqrt{3}}{2} + 1 \cdot \frac{1}{2} = -\sqrt{3} + \frac{1}{2}\]\[y' = 2 \cdot \frac{1}{2} + 1 \cdot \frac{\sqrt{3}}{2} = 1 + \frac{\sqrt{3}}{2}\]

Step 6: Final Calculation

Simplify the final results:\[x' \approx -1.732 + 0.5 = -1.232\]\[y' \approx 1 + 0.866 = 1.866\]

Step 7: Conclusion

The point \((-2,1)\) rotates to \((-1.232, 1.866)\) after a rotation of \(30^{\circ}\).

Key Concepts

Trigonometric Values

Understanding trigonometric values is essential when dealing with rotation transformations. Trigonometric functions such as sine and cosine relate the angles of a triangle to the lengths of its sides. Specifically, these values are crucial because they are used in formulas to determine the new positions of points after a rotation. For a 30-degree angle, which is a common angle in trigonometry, the values are straightforward:

• For \( \cos(30^{\circ}) \), the value is \(\frac{\sqrt{3}}{2}\).
• For \( \sin(30^{\circ}) \), the value is \(\frac{1}{2}\).

These trigonometric values are derived from the properties of 30-degree angles in the unit circle and help make precise calculations during rotations on the coordinate plane.

Coordinate Axes

The coordinate axes, namely the x-axis and y-axis, form the foundational framework of any coordinate plane. Each point on this plane is described using a pair of \( (x, y) \) values. When you're dealing with questions involving rotations, it's the axes themselves that shift, rather than the points.
This concept can initially be tricky because it requires you to visualize or calculate the position of points from a new set of axes. Imagine the whole grid of the coordinate plane rotates while the point remains fixed to its grid location—only its relative coordinates change. This requires applying rotation formulas to determine new coordinates:

• New x-coordinate: \( x' = x \cos\phi + y \sin\phi \)
• New y-coordinate: \( y' = -x \sin\phi + y \cos\phi \)

These formulas help in calculating where the axes' rotation places the point on the rotated system.

Origin Rotation

Rotation about the origin is a transformation that involves turning the whole coordinate system around the origin point \( (0, 0) \). The origin acts as a pivot point. In this type of rotation, although the visual position of objects like points appears unchanged regarding their spot on the plane, their coordinates in reference to the new axes change.
For example, when the problem asks you to rotate \(30^{\circ}\) around the origin, it means every point will shift in such a way that if you were looking from above, it would seem as if the axes themselves have turned by 30 degrees. This kind of rotation requires:

• Referring to trigonometric values to calculate the new coordinate locations.
• Applying transformation formulas specifically tailored for angle rotations, such as those for sine and cosine functions.

Keep in mind that the physical location on the paper or plane does not change, only the frame of reference for the point does. This different perspective is why it's called rotation about the origin.