Problem 51 - Algebra and Trigonometry

Question

\(51-54\). Evaluate each expression under the given conditions. $$ \cos 2 \theta ; \sin \theta=-\frac{3}{5}, \theta \text { in Quadrant III } $$

Step-by-Step Solution

Answer

\( \cos 2\theta = \frac{7}{25} \)

Step 1: Determine the cosine of theta

The trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \) can be used to find \( \cos \theta \). Since \( \sin \theta = -\frac{3}{5} \), we square this to get \( \sin^2 \theta = \left(-\frac{3}{5}\right)^2 = \frac{9}{25} \). Then \( \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \). Therefore, \( \cos \theta = \pm \frac{4}{5} \). In Quadrant III, cosine is negative, so \( \cos \theta = -\frac{4}{5} \).

Step 2: Apply the double angle formula for cosine

The double angle formula for cosine is \( \cos 2\theta = 2\cos^2 \theta - 1 \). Using the value from Step 1, \( \cos \theta = -\frac{4}{5} \), we calculate \( \cos^2 \theta = \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \).

Step 3: Calculate cosine of 2 theta

Substitute \( \cos^2 \theta = \frac{16}{25} \) into the double angle formula: \( \cos 2\theta = 2 \times \frac{16}{25} - 1 \). This simplifies to \( \frac{32}{25} - 1 = \frac{32}{25} - \frac{25}{25} = \frac{7}{25} \).

Key Concepts

Double Angle Formula

The double angle formula is a valuable tool in trigonometry that helps simplify expressions involving angles that are double of a certain angle. It allows us to express trigonometric functions of twice an angle in terms of trigonometric functions of the original angle. In the context of this problem, we are focusing on the formula for cosine, which is:

• \( \cos 2\theta = 2\cos^2 \theta - 1 \)

This formula can often be derived by using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \).
Using the double angle formula can make it easier to resolve problems when angles are doubled, and it simplifies calculations. It can also be rewritten using sine if necessary:

• \( \cos 2\theta = 1 - 2\sin^2 \theta \)

By using this formula, you can clearly see how the properties of an angle relate to its double, helping to reveal interesting trigonometric relationships.

Quadrants of the Unit Circle

To better understand trigonometric values, we must understand the unit circle's quadrants. The unit circle is divided into four quadrants, and each quadrant determines the signs of trigonometric functions. The problem specifies that \( \theta \) is in Quadrant III.
Here's a quick guide to the unit circle quadrants:

• **Quadrant I**: Both sine and cosine are positive.
• **Quadrant II**: Sine is positive; cosine is negative.
• **Quadrant III**: Both sine and cosine are negative.
• **Quadrant IV**: Cosine is positive; sine is negative.

Knowing this, if \( \sin \theta = -\frac{3}{5} \) in Quadrant III, the cosine must also be negative, reinforcing that \( \cos \theta = -\frac{4}{5} \). This understanding of the unit circle's quadrants is crucial for solving trigonometric problems as it dictates the sign of the result.

Cosine and Sine Relationship

The relationship between cosine and sine is fundamental in trigonometry. One of the key relationships is the Pythagorean identity:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity illustrates that sine and cosine functions are interconnected. By knowing one of these values, you can always calculate the other, given the quadrant information. In this example, since \( \sin \theta = -\frac{3}{5} \), we used the identity to find \( \cos \theta \) by simply rearranging the identity:\( \cos^2 \theta = 1 - \sin^2 \theta \).
By substituting the sine value, we find that \( \cos^2 \theta = \frac{16}{25} \). Including quadrant information (that cosine is negative in Quadrant III), we determine that \( \cos \theta = -\frac{4}{5} \).This fundamental relationship allows for seamless transitions between sine and cosine functions, especially when working with reflected angles on the unit circle. Understanding these connections is critical for mastering trigonometry.