Problem 60 - Precalculus : Building Concepts and Connections

Question

Write the polar equation \(r=4 \sin \left(\theta-\frac{3 \pi}{2}\right)\) in terms of just the cosine function.

Step-by-Step Solution

Answer

The polar equation \(r=4 \sin \left(\theta-\frac{3 \pi}{2}\right)\) simplifies to \(r=4 \cos(\theta)\) when written in terms of the cosine function.

Step 1: Recall Relationship Between Sine and Cosine

One of the main trigonometric identities we have is the phase shift between sine and cosine, which states \(\cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right)\). This means that we can replace sine by cosine with a shift of \(\frac{\pi}{2}\) to the right.

Step 2: Utilize the Relationship to Rewrite the Given Equation

Use the phase-shift relationship between sine and cosine for the given equation. Rename the existing phase shift \(\theta - \frac{3\pi}{2}\) into another variable, say \(\phi\). Therefore, we have \(\sin(\phi) = \cos\left(\phi-\frac{\pi}{2}\right)\). Now, replace \(\sin(\phi)\) by \(\cos\left(\phi - \frac{\pi}{2}\right)\) in the original equation. Hence, the original equation \(r=4 \sin(\phi)\) becomes \(r=4 \cos\left(\phi-\frac{\pi}{2}\right)\).

Step 3: Substitute Back the Original Variable

Finally, revert back from \(\phi\) to \(\theta\). Substitute \(\phi = \theta - \frac{3\pi}{2}\) back into \(r=4 \cos\left(\phi-\frac{\pi}{2}\right)\). This results in \(r=4 \cos\left(\left(\theta - \frac{3\pi}{2}\right) - \frac{\pi}{2}\right)\), which simplifies to \(r=4 \cos\left(\theta - 2\pi\right)\). The period of cosine function is \(2\pi\), so the phase shift of \(2\pi\) or any multiple of it doesn't affect the function, hence, the final equation simplifies to \(r=4 \cos(\theta)\).

Key Concepts

Trigonometric identities

Trigonometric identities are essential tools in trigonometry that show relationships between different trigonometric functions. One common identity is the relationship between the sine and cosine functions. The identity \(\cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right)\\) demonstrates a phase shift between sine and cosine. This is akin to shifting the sine wave by \(\frac{\pi}{2}\) radians to obtain the cosine wave.
This identity is vital when working with polar equations because it allows one function to be rewritten in terms of the other. This technique can simplify calculations or tailor equations to specific requirements in trigonometry.
Applying this concept helps in solving exercises such as converting sine-based polar equations into cosine-based equations, highlighting the significance of understanding these foundational identities.

Cosine function

The cosine function is a fundamental trigonometric function represented as \(\cos(\theta)\). It forms the horizontal component of the unit circle and has a wide range of applications in mathematics and physics. The cosine function describes the relationship between an angle in radians and the x-coordinate on the unit circle.
Key characteristics of the cosine function include:

• Amplitudes and period: The standard cosine function oscillates between -1 and 1, with an amplitude of 1 and a period of \(2\pi\).
• Symmetry: It is an even function, meaning that \(\cos(-\theta) = \cos(\theta)\).
• Phase shifts: These allow for adjustments to the cosine graph by adding or subtracting a constant angle.

Understanding the cosine function's properties is crucial for solving polar equations, as demonstrated by rewriting a polar equation from sine terms to cosine terms. Recognizing how to manipulate these properties allows for efficient equation transformation and problem-solving.

Phase shift

A phase shift in trigonometry refers to the horizontal movement of a trigonometric function graph along the x-axis. This concept is essential when transforming functions, especially in equations involving sine and cosine.
To incorporate a phase shift, you add or subtract a constant angle from the variable \(\theta\). For example:

• A positive phase shift, \(\theta + a\), moves the graph to the left.
• A negative phase shift, \(\theta - a\), moves the graph to the right.

In the given exercise, the equation \(r = 4 \sin\left(\theta - \frac{3\pi}{2}\right)\) requires rewriting in terms of only cosine functions. Utilizing phase shift helps convert this equation by identifying the sine function's equivalent cosine form. By acknowledging the phase shift \(-\frac{3\pi}{2}\) and understanding the identity \(\sin(\theta) = \cos\left(\theta - \frac{\pi}{2}\right)\), we rewrite the equation as \(r = 4 \cos(\theta)\). Grasping these shifts is vital, not just for rewriting polar equations but also for visualizing transformations on trigonometric graphs.