Problem 59 - Precalculus : Building Concepts and Connections

Question

Write the polar equation \(r=2-2 \cos \left(\theta+\frac{\pi}{2}\right)\) in terms of just the sine function.

Step-by-Step Solution

Answer

The polar equation written in terms of the sine function is \(r = 2+ 2 \sin \theta \)

Step 1: Identify the Key Trigonometric Identity

Use the mathematical identity \( \sin(x) = \cos\left(\frac{\pi}{2} - x\right)\) to change the \(\cos\) function in our equation into a \(\sin\) function.

Step 2: Apply the Trigonometric Identity

Replace \(\cos\) in the original equation with the expression above: \[r=2-2 \cos \left(\theta+\frac{\pi}{2}\right) = 2 - 2\sin\left(\frac{\pi}{2} - \left(\theta+\frac{\pi}{2}\right)\right)\] Simplify the second part in the brackets \[\frac{\pi}{2} - \left(\theta+\frac{\pi}{2}\right) =\frac{\pi}{2}-\theta-\frac{\pi}{2} = -\theta\] So the equation becomes \[r=2- 2 \sin (-\theta)\]

Step 3: Simplify Further Due to Trigonometric Property

As we know that \(\sin(-\theta) = -\sin(\theta)\), substitute \(\sin(-\theta)\) in the equation with \(-\sin(\theta)\), gives: \[ r = 2+ 2 \sin \theta \]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. They play a vital role in simplifying expressions and solving equations involving trigonometric functions like sine, cosine, and tangent. A fundamental identity used in polar equations is the co-function identity, which relates sine and cosine functions. The identity \( \sin(x) = \cos\left(\frac{\pi}{2} - x\right) \) comes in handy when you need to swap a cosine function for a sine function.

Understanding the co-function identity lets us transition between different trigonometric functions, making it easier to work with equations that might seem challenging at first glance. For instance, this identity allows us to convert the polar equation \( r = 2 - 2 \cos\left(\theta + \frac{\pi}{2}\right) \) into a form with only the sine function present, which can be easier to analyze or graph in certain contexts.

Sine Function

The sine function, denoted as \( \sin \), is one of the primary trigonometric functions and describes the ratio of the opposite side to the hypotenuse in a right-angled triangle. Sine's graph is a wave that oscillates between -1 and 1, which makes it periodic with a period of \( 2\pi \). This wave-like behavior is reflected in numerous physical phenomena, such as sound waves and tides.

When dealing with polar equations, the sine function aids in representing circular movements and patterns. For example, the sine function helps describe the radius of a point in a polar coordinate system, as shown in the equation \( r = 2 + 2 \sin \theta \), which represents a circle that has been transformed by a sine function. In practice, understanding the characteristics of the sine function, such as its amplitude, period, and phase shift, allows students to graph and interpret trigonometric equations more effectively.

Converting Cosine to Sine

Converting cosine to sine involves using trigonometric identities, particularly the co-function identities mentioned earlier. These identities are based on the complementary relationship between sine and cosine. In the context of Cartesian coordinates, a point's x-coordinate corresponds to the cosine function value, and its y-coordinate corresponds to the sine function value at a given angle.

The conversion process helps simplify equations or change their form to reveal insights that may not be as apparent in their original state. For instance, by converting a cosine function to a sine function, one might adjust an equation to a preferred trigonometric function for further analysis, graphing or integration into other mathematical models. The co-function identity \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta ) \) serves as a bridge between these two fundamental trigonometric functions, enhancing versatility and understanding in solving trigonometric problems.