Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of \(\sin 2 \theta, \cos 2 \theta\), and \(\tan 2 \theta\) for the given values of \(\theta\). $$ \cot \theta=\frac{4}{3} ; \quad 180^{\circ}<\theta<270^{\circ} $$

Step-by-Step Solution

Answer

\( \sin 2\theta = \frac{24}{25} \), \( \cos 2\theta = \frac{7}{25} \), \( \tan 2\theta = \frac{24}{7} \).

Step 1: Determine Quadrant for \( \theta \)

Given that \( 180^{\circ} < \theta < 270^{\circ} \), \( \theta \) is in the third quadrant, where both sine and cosine are negative.

Step 2: Use Identity to Find \( \tan \theta \)

We know that \( \cot \theta = \frac{4}{3} \). Since \( \cot \theta = \frac{1}{\tan \theta} \), it follows that \( \tan \theta = \frac{3}{4} \). In the third quadrant, \( \tan \theta \) is positive.

Step 3: Solve for \( \sin \theta \) and \( \cos \theta \) Using \( \tan \theta \)

From \( \tan \theta = \frac{3}{4} \), we can let \( y = 3 \) and \( x = 4 \), giving us a reference triangle with opposite side 3, adjacent side 4, and hypotenuse \( \sqrt{3^2 + 4^2} = 5 \). Hence, \( \sin \theta = -\frac{3}{5} \) and \( \cos \theta = -\frac{4}{5} \) since both sine and cosine are negative in the third quadrant.

Step 4: Calculate \( \sin 2\theta \) Using Double Angle Formula

Use the formula \( \sin 2\theta = 2 \sin \theta \cos \theta \).Substituting, we have \( \sin 2\theta = 2 \cdot \left( -\frac{3}{5} \right) \cdot \left( -\frac{4}{5} \right) = \frac{24}{25} \).

Step 5: Calculate \( \cos 2\theta \) Using Double Angle Formula

Use the formula \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).Substitute for \( \cos \theta = -\frac{4}{5} \) and \( \sin \theta = -\frac{3}{5} \) to get \( \cos 2\theta = \left(-\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} \).

Step 6: Calculate \( \tan 2\theta \) Using Double Angle Formula

Use the formula \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).Substitute \( \tan \theta = \frac{3}{4} \) to get \( \tan 2\theta = \frac{2 \cdot \frac{3}{4}}{1 - \left( \frac{3}{4} \right)^2} = \frac{\frac{6}{4}}{1 - \frac{9}{16}} = \frac{\frac{3}{2}}{\frac{7}{16}} = \frac{24}{7} \).

Key Concepts

Double Angle Formulas

Double angle formulas are pivotal in trigonometry. They let us express trigonometric functions of double angles, like \(2\theta\), in terms of trigonometric functions of \(\theta\). The formulas for sine, cosine, and tangent of double angles help simplify computations and solve equations.

Here are the primary double angle identities you should know:

• For sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• For cosine: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• For tangent: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \)

These identities are derived from fundamental trigonometric identities. They simplify working with angles by linking complex expressions to basic functions. For instance, knowing \( \sin \theta \) and \( \cos \theta \), you can easily calculate \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \) without directly measuring double angle values.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are crucial for studying angles and modeling periodic phenomena. They are based on the right triangle model. Given a right triangle:

• \( \sin \theta = \frac{\text{opposite}\: \text{side}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent}\: \text{side}}{\text{hypotenuse}} \)
• \( \tan \theta = \frac{\text{opposite}\: \text{side}}{\text{adjacent}\: \text{side}} \)

These functions serve as the foundation for trigonometric identities and relationships.

Each function has a specific range of values depending on the angle. For instance, sine and cosine values run from \(-1\) to \(1\), showing their periodic nature. Moreover, understanding these functions helps with various applications, such as calculating the angles and side lengths in triangles. In this exercise, these relationships helped determine \( \sin \theta \) and \( \cos \theta \) from \( \tan \theta \), using known values.

Quadrant Analysis

Quadrant analysis helps determine the sign and value range for trigonometric functions. This understanding is vital when dealing with angles beyond the first 90 degrees.The coordinate plane is divided into four 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: Sine is negative; cosine is positive.

In this exercise, we considered \( \theta \) in the third quadrant where both sine and cosine are negative. This factor is crucial as it helps in determining the correct signs of \( \sin \theta \) and \( \cos \theta \). Proper quadrant analysis prevents mistakes in calculation, ensuring the angles' trigonometric functions are applied correctly based on their sign and magnitude.