Problem 59 - College Trigonometry

Question

60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for … # Convert the equation from rectangular coordinates into polar coordinates. Solve for \)r\( in all but \)\\# 60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for \)\theta$. $$ y=7 $$

Step-by-Step Solution

Answer

The polar equation is \( r\sin(\theta) = 7 \), where \( \theta = \sin^{-1}(\frac{7}{r}) \).

Step 1: Understand the Given Equation

The equation provided is in rectangular coordinates, written as \( y = 7 \). This means that the y-coordinate is constantly 7 no matter what the x-coordinate is.

Step 2: Recall Conversion Formulas

To convert from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use two primary formulas: \( r = \sqrt{x^2 + y^2} \) and \( \tan(\theta) = \frac{y}{x} \).

Step 3: Set Up Equation Using Given Information

Since \( y = 7 \), we relate this to the polar form by using \( y = r\sin(\theta) \). Therefore, we have \( r\sin(\theta) = 7 \).

Step 4: Solve for \(\theta\)

As instructed not to solve for \( r \), but for \( \theta \), we consider \( \theta \) from the basic identity \( \sin(\theta) = \frac{y}{r} \). From our setup, \( \sin(\theta) = \frac{7}{r} \). To find \( \theta \), rely on \( \theta = \sin^{-1}(\frac{7}{r}) \).

Step 5: Consider Special Quadrants

Interpret \( y = 7 \) on the Cartesian plane: it represents a horizontal line parallel to the x-axis at y = 7. Thus, on a polar grid, this line is a circle that cuts through points at \( y = 7 \) repeatedly for all x-values. This implies potential angles (\( \theta \)) based on the graph context.

Step 6: State Result in Polar Coordinates

The polar representation satisfies \( r\sin(\theta) = 7 \) for variations of \( \theta = \sin^{-1}(\frac{7}{r}) \), indicating multiple angles that fulfill the condition throughout the circle where possible.

Key Concepts

Conversion Formulas

Converting equations between coordinate systems is a foundational skill in mathematics, especially useful in calculus and trigonometry. To convert between rectangular and polar coordinates, we apply specific formulas that bridge these two systems:

• From Rectangular to Polar: Use the distance formula to find the radius: \( r = \sqrt{x^2 + y^2} \). This equation calculates the distance of the point from the origin. For the angle, use the formula \( \tan(\theta) = \frac{y}{x} \) to determine the angle relative to the positive x-axis.
• From Polar to Rectangular: Convert back to the familiar x and y using \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \).

These formulas are essential for moving fluidly between the two coordinate systems, offering flexibility depending on the needs of the problem.
Polar coordinates emphasize the magnitude and direction, while rectangular coordinates focus on direct horizontal and vertical distances.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a location on a plane using an ordered pair \((x, y)\). Each point on this system specifies how far horizontally (x) and vertically (y) you need to move from the origin to reach it.

• This system is particularly useful for algebraic manipulation, allowing straightforward addition, subtraction, and interpretation of linear equations like \( y = mx + b \).
• They are advantageous when tackling problems involving integrals and derivatives or when the equations are linear.

In the exercise \( y = 7 \), the rectangular coordinate system is used initially, demonstrating that regardless of x, y remains constant at 7.
This results in a horizontal line on the graph, aligning parallel to the x-axis for all x-values.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. These identities are invaluable when resolving equations within and across coordinate systems:

• Sine and Cosine Functions: They represent y and x values in a unit circle framework. For any angle \( \theta \), \( \sin(\theta) \) provides the y-coordinate of the point on the circle, while \( \cos(\theta) \) gives the x-coordinate.
• Inverse Trigonometric Functions: Given a trigonometric value, these functions help determine the corresponding angle \( \theta \). In the exercise, \( \theta = \sin^{-1}(\frac{7}{r}) \) specifies multiple angles satisfying the condition when y is consistent, like in \( y = 7 \).

These identities help in converting and solving scenarios involving multiple coordinate systems, facilitating easier transformations between functions.

Angle Determination

Determining angles in polar coordinates involves locating where a point lies relative to the polar axis, which extends horizontally from the origin. This process often requires the use of trigonometric identities:

• For a line parallel to the x-axis, like \( y = 7 \), angles could vary significantly since it extends indefinitely. However, using \( \tan(\theta) = \frac{y}{x} \), each point on the line needs analysis depending on its x-coordinate.
• In this scenario, the specific angle values are derived using \( \theta = \sin^{-1}(\frac{7}{r}) \). This results from recognizing potential intersection angles of the specified y value intersecting the polar plot.

Understanding polar coordinates and angle determination aids in visualizing complex relationships.
It clarifies how a constant y affects varying x, demonstrating the many possible \( \theta \) values fitting the description on a polar graph.