Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Consider the graph of \(r=f(\sin \theta)\). (a) Show that when the graph is rotated counterclockwise \(\pi / 2\) radians about the pole, the equation of the rotated graph is \(r=f(-\cos \theta)\). (b) Show that when the graph is rotated counterclockwise \(\pi\) radians about the pole, the equation of the rotated graph is \(r=f(-\sin \theta)\). (c) Show that when the graph is rotated counterclockwise \(3 \pi / 2\) radians about the pole, the equation of the rotated graph is \(r=f(\cos \theta)\).

Step-by-Step Solution

Answer

The equations of the rotated graph for \(r=f(\sin \theta )\) are \(r=f(\cos \theta )\) after rotation by \(\pi / 2\) radians, \(r=f(-\sin \theta)\) after rotation by \(\pi\) radians, and \(r=f(-\cos \theta)\) after rotation by \(3 \pi / 2\) radians.

Step 1: Rotate the graph by \(\pi / 2\) radians

Rotating the graph counterclockwise by \(\pi / 2\) radians amounts to replacing \(\theta\) with \(\theta - \pi / 2\). Thus, the equation of the rotated graph is \(r=f(\sin (\theta - \pi / 2))\). Now, using the identity \(\sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\), with \(\alpha=\theta\) and \(\beta=\pi / 2\), this gives \(r=f(\cos \theta)\), as \(\sin \pi / 2 = 1\) and \(\cos \pi / 2 = 0\). Therefore, this new equation represents the graph when it is rotated counterclockwise by \( \pi / 2\) radians about the pole.

Step 2: Rotate the graph by \(\pi\) radians

Rotating the graph counterclockwise by \(\pi\) radians amounts to replacing \(\theta\) with \(\theta - \pi\). Thus, the equation of the rotated graph is \(r=f(\sin (\theta - \pi))\). Here, using the same trigonometric identity, we get \(r=f(-\sin \theta)\), as \(\sin \pi = 0\) and \(\cos \pi = -1\). This new equation represents the graph when it is rotated counterclockwise by \( \pi\) radians about the pole.

Step 3: Rotate the graph by \(3 \pi / 2\) radians

Finally, rotating the graph counterclockwise by \(3 \pi / 2\) radians amounts to replacing \(\theta\) with \(\theta - 3 \pi / 2\). The equation of the rotated graph then becomes \(r=f(\sin (\theta - 3 \pi / 2))\). Using the relevant trigonometric identity again we get \(r=f(-\cos \theta)\), as \(\sin 3 \pi / 2 = -1\) and \(\cos 3 \pi / 2 = 0\). This new equation represents the graph when it is rotated counterclockwise by \( 3 \pi / 2\) radians about the pole.

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools used to simplify and solve problems involving angles and sides of triangles. They relate the angles of a triangle to the ratios of its sides, allowing for transformations and simplifications of expressions. In polar coordinates, these identities help in transforming polar equations through rotations or shifts.

For example, common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle Addition Identities: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• Angle Subtraction Identities: \( \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)

These identities are particularly handy when dealing with rotations. For instance, in the solution of the original exercise, the identity \( \sin(\theta - \frac{\pi}{2}) = \cos \theta \) is used to transform the equation \( r = f(\sin \theta) \) into \( r = f(-\cos \theta) \), showing how the graph rotates a specific angle about the pole.

Graph Rotation

Understanding how graph rotation works is essential when dealing with polar equations. Rotating a graph in polar coordinates involves changing the angle \( \theta \) according to the direction and degree of rotation. This is particularly important in exercises where you need to visualize the placement of a graph after being rotated.

To rotate a graph counterclockwise:

• By \( \pi / 2 \) radians, you replace \( \theta \) with \( \theta - \pi / 2 \). This means the sine in the original equation \( r=f(\sin \theta) \) should be transformed using trigonometric identities, ending up as \( r=f(-\cos \theta) \).
• For \( \pi \) radians, replace \( \theta \) with \( \theta - \pi \). The rotation affects your equation to become \( r=f(-\sin \theta) \).
• When you rotate by \( 3\pi / 2 \) radians, \( \theta \) becomes \( \theta - 3\pi / 2 \), resulting in \( r=f(\cos \theta) \).

Graph rotation is a geometric transformation that helps to visualize data and understand symmetry and alignment of polar plots. Knowing how to rotate graphs accurately allows for clearer comprehension of how alterations to \( \theta \) affect the overall graph structure.

Polar Equations

Polar equations represent points in the plane using a distance from a reference point and an angle from a reference direction. These equations offer a different perspective than Cartesian coordinates, dealing with curves that are esoteric or difficult to handle using standard linear equations.

Polar coordinates consist of:

• The radial coordinate \( r \), which is the length from the pole (origin) to the point.
• The angular coordinate \( \theta \), which is the angle from the positive x-axis to the line connecting the pole with the point.

In exercises like the original, polar equations are manipulated through trigonometric identities and graph rotations to study their properties. Understanding how to adjust the angle \( \theta \) allows for modifications in the graph's visual orientation. For instance, if you rotate the graph, you adjust \( \theta \) and use identities to interpret new equations, providing insights into the symmetry and periodic nature of the resulting design.

By mastering polar equations, students can solve complex problems in physics and engineering, leveraging the unique benefits of this coordinate system for circular and spiral motion representation and beyond.