Problem 29 - Thomas Calculus

Question

Show that the point \((2,3 \pi / 4)\) lies on the curve \(r=2 \sin 2 \theta\)

Step-by-Step Solution

Answer

Yes, the point \((2, \frac{3\pi}{4})\) lies on the curve. Polar coordinates \((2, \frac{3\pi}{4})\) and \((-2, -\frac{\pi}{4})\) represent the same point.

Step 1: Verify Polar Coordinates

We are given the point in polar coordinates as \((r, \theta) = (2, \frac{3\pi}{4})\). To show that this point lies on the curve, we need to check whether this point satisfies the equation of the curve given by \(r = 2 \sin 2\theta\).

Step 2: Substitute Theta into the Curve Equation

Substitute \(\theta = \frac{3\pi}{4}\) into the expression for \(r\) on the curve. This gives us \(r = 2 \sin \left(2 \times \frac{3\pi}{4}\right)\).

Step 3: Compute Sin of Double Angle

Calculate \(2 \theta = 2 \times \frac{3\pi}{4} = \frac{3\pi}{2}\). The sine of \(\frac{3\pi}{2}\) is \(-1\).

Step 4: Solve for r

Now evaluate \(r = 2 \sin \left(\frac{3\pi}{2}\right) = 2 \times (-1) = -2\).

Step 5: Compare r Values

The given point has \(r = 2\), but when \(\theta = \frac{3\pi}{4}\), the curve gives \(r = -2\). In polar coordinates, these points are equivalent because both represent the same point in the polar plane but with different directions.

Key Concepts

Curve Equation

In polar coordinates, a curve equation describes the relationship between the radius \(r\) and the angle \(\theta\). Instead of identifying points based on \((x, y)\) coordinates as in the Cartesian system, polar coordinates specify each point by how far it is from the origin (the pole) and the angle it makes with the positive x-axis.

For example, the curve given by \(r = 2 \sin 2\theta\) implies that the radius changes based on the sine of double the angle, \(2\theta\). This equation helps us generate a specific curve by defining all the points \((r, \theta)\) that satisfy it.

To determine if a point lies on this curve, substitute \(\theta\) from the point into the equation to compute its radius. If the calculated \(r\) matches the one given in the polar coordinates of the point, then the point indeed lies on the curve. In polar coordinates, a mismatch in "direction" is allowable if it results in analogous points, which can happen due to the circular nature of polar graphs.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are vital in linking angles to ratios of a triangle's sides, or in this case, angles to distances in polar coordinates. The sine function \(\sin\theta\) provides values that oscillate between -1 and 1, and it is periodic with a cycle of \(2\pi\), which means it repeats its values in every full rotation.

In the given exercise, the sine function is evaluated at a double angle, which is part of the analysis of the curve \(r = 2 \sin 2\theta\). Understanding the behavior of \(\sin\theta\) with respect to different angles \(\theta\) can tell us how the radius of a point changes as \(\theta\) varies. By analyzing trigonometric values, we can explore how points align with certain curves, especially when the function is part of the curve's formula.

Double Angle Identity

The double angle identity is a trigonometric formula used to simplify expressions involving trigonometric functions of doubled angles. For sine, it is given by \(\sin 2\theta = 2\sin\theta\cos\theta\), but in this exercise, the actual function 'inside' the formula was used directly rather than expressed through the identity.

The point of this problem involved evaluating the sine at \(2\theta\). First, multiply the given angle \(\theta = \frac{3\pi}{4}\) by 2 to get \(\frac{3\pi}{2}\). Then, using the sine function, \(\sin \frac{3\pi}{2} = -1\), which leads to finding the value of \(r\) on the curve.

This identity aids in understanding how changes in angle affect calculations in polar coordinate systems. It can also be adapted and used to solve more complex problems involving trigonometric functions by making expressions more manageable.