Problem 35 - Algebra 2

Question

Given \(\cos \theta=\frac{3}{5} \operatorname{and} 270^{\circ}<\theta<360^{\circ},\) find the exact value of each expression. $$ \cos 2 \theta $$

Step-by-Step Solution

Answer

The exact value of \( cos2\theta \) is \( \frac{11}{25} \).

Step 1: Find sin θ

We need to find \(sin \theta \). We know that \( sin^2 \theta = 1 - \cos^2 \theta \). Using the given \( cos \theta = \frac{3}{5} \), we get \( sin^2 \theta = 1 - (\frac{3}{5})^2 = 1 - \frac{9}{25} = \frac{16}{25} \). Therefore, \( sin \theta = \pm \frac{4}{5} \). However, since \( 270^{\circ} < \theta < 360^{\circ} \), we are in the 4th quadrant where sine is negative, so \( sin \theta = - \frac{4}{5} \).

Step 2: Apply the double angle formula

The double angle formula for cosine is \( cos2\theta = 2\cos^2 \theta - 1 \). Substituting \( \cos\theta = \frac{3}{5} \) into the formula, we get \( cos2\theta = 2(\frac{3}{5})^2 - 1 = 2*\frac{9}{25} - 1 = \frac{36}{25} - 1 = \frac{11}{25} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental expressions that hold true for all angle values and play a vital role in simplifying and solving trigonometric problems. They help link different trigonometric functions and allow us to transform complex expressions into simpler forms. In our example, the double angle formula for cosine is one such identity:

• The formula is given by: \( \ \cos 2\theta = 2\cos^2 \theta - 1 \).
• This formula is derived from the Pythagorean identity and simplifies calculating the cosine of double angles when the initial angle is known.
• Other related trigonometric identities include the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) and the angle addition formulas.

These identities form the backbone of trigonometry, providing us the tools to manipulate and solve equations involving trigonometric functions effectively.

Cosine Function

The cosine function, one of the primary trigonometric functions, reflects the x-coordinate of a point on the unit circle corresponding to a given angle. It is defined for all real numbers and has a periodicity of \(360^\circ\) (or \(2\pi\)):

• The cosine value ranges from \(-1\) to \(1\) and is crucial in determining the horizontal component of vectors, waves, and oscillations.
• It is often used in real-world applications, such as calculating distances and determining phase shifts in waves.

Understanding its properties through identities, like the double angle formula, allows us to easily find cosine values for compound angles, enabling more efficient solutions to trigonometric equations. For instance, knowing \( \cos \theta = \frac{3}{5} \), we can readily apply the double-angle formula to find \( \cos 2\theta \).

Quadrants in Trigonometry

Trigonometry divides the coordinate plane into four quadrants, each representing different sign rules for trigonometric functions:

• The first quadrant (\(0^\circ\) to \(90^\circ\)) has all trigonometric functions positive.
• The second quadrant (\(90^\circ\) to \(180^\circ\)) has sine positive, while cosine and tangent are negative.
• The third quadrant (\(180^\circ\) to \(270^\circ\)) has tangent positive, with sine and cosine negative.
• The fourth quadrant (\(270^\circ\) to \(360^\circ\)) has cosine positive, and sine and tangent negative.

In our problem, we are given \(270^\circ < \theta < 360^\circ\), placing \(\theta\) in the fourth quadrant. Here, the cosine function remains positive, but sine is negative, as illustrated by \( \sin \theta = -\frac{4}{5} \). Understanding these quadrant rules is crucial for accurately solving trigonometric problems involving angles in different parts of the coordinate plane.