Problem 2 - Precalculus: Functions and Graphs

Question

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

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: Identify the Quadrant

The angle \(\theta\) is between \(180^\circ\) and \(270^\circ\). This means that \(\theta\) is in the third quadrant, where both sine and cosine functions are negative.

Step 2: Find Sine and Cosine of Theta

Given \(\cot \theta = \frac{4}{3}\), we know that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Therefore, \(\cos \theta = 4k\) and \(\sin \theta = 3k\) for some constant \(k\). Using the Pythagorean identity, \(\sin^2 \theta + \cos^2 \theta = 1\), we solve for \(k\).\[ (3k)^2 + (4k)^2 = 1 \] \[ 9k^2 + 16k^2 = 1 \] \[ 25k^2 = 1 \] \[ k^2 = \frac{1}{25} \] \[ k = \frac{1}{5} \] Thus, \(\sin \theta = -\frac{3}{5}\) and \(\cos \theta = -\frac{4}{5}\), considering their signs in the third quadrant.

Step 3: Calculate \(\sin 2\theta\)

Use the double angle formula for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\). Substitute the values found: \[ \sin 2\theta = 2 \left(-\frac{3}{5}\right) \left(-\frac{4}{5}\right) = 2 \cdot \frac{12}{25} = \frac{24}{25} \] So, \(\sin 2\theta = \frac{24}{25}\).

Step 4: Calculate \(\cos 2\theta\)

Use the double angle formula for cosine: \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\). Substitute the values: \[ \cos 2\theta = \left(-\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} \] So, \(\cos 2\theta = \frac{7}{25}\).

Step 5: Calculate \(\tan 2\theta\)

Use the relationship \(\tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta}\). Using the values calculated: \[ \tan 2\theta = \frac{\frac{24}{25}}{\frac{7}{25}} = \frac{24}{7} \] Therefore, \(\tan 2\theta = \frac{24}{7}\).

Key Concepts

Double Angle Formulas

Double angle formulas are incredibly handy when solving trigonometric problems that involve expressions like \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \). These formulas give us a way to express these double angles using the sine and cosine of a single angle. Here’s a quick look at each:

• \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• \( \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \)

For example, if you know the sine and cosine of an angle, you can readily compute the sine and cosine of its double angle. For the angle \( \theta \), knowing \( \sin \theta \) and \( \cos \theta \) not only allows us to find \( \sin 2\theta \) and \( \cos 2\theta \) but also helps in solving other related expressions by employing these formulas.
Using these formulas, we were able to calculate \( \sin 2\theta \) as \( \frac{24}{25} \), \( \cos 2\theta \) as \( \frac{7}{25} \), and in turn \( \tan 2\theta \) as \( \frac{24}{7} \). Understanding and applying these formulas effectively is fundamental in trigonometry!

Pythagorean Identity

The Pythagorean identity is a crucial concept in trigonometry, and it's similar to the geometric theorem it's named after. The formula \( \sin^2 \theta + \cos^2 \theta = 1 \) is used to relate the sine and cosine of an angle to 1, which heavily relates to the unit circle.
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. Every point on this circle can be represented as \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle formed with the positive x-axis. This makes it a powerful tool for finding missing values, especially when dealing with trigonometric identities.
In our problem, once we had \( \cot \theta = \frac{4}{3} \), we set \( \cos \theta = 4k \) and \( \sin \theta = 3k \). Employing the Pythagorean identity enabled us to solve for the scaling factor \( k \), eventually determining that \( \sin \theta = -\frac{3}{5} \) and \( \cos \theta = -\frac{4}{5} \). This step is crucial, particularly because it underpins our ability to use double angle formulas accurately.

Third Quadrant Angles

In trigonometry, understanding which quadrant an angle lies in is essential because it affects the sign of the sine, cosine, and tangent values.

Angles in the third quadrant (between \(180^\circ\) and \(270^\circ\)) have unique characteristics:

• Both sine \( \sin \theta \) and cosine \( \cos \theta \) are negative.
• The tangent \( \tan \theta \), which is the ratio \( \frac{\sin \theta}{\cos \theta} \), is positive because a negative divided by a negative is positive.

When computing \( \sin \theta \) and \( \cos \theta \) for angles within this range, one must ensure the correct sign is considered. These correct signs ensure the consistency and correctness when further applying formulas like the double angle identities.
In our example, recognizing that \( \theta \) is in the third quadrant led us to adjust \( \sin \theta = -\frac{3}{5} \) and \( \cos \theta = -\frac{4}{5} \), which subsequently guided us to accurate results for \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \). Always be mindful of which quadrant the angle is in when solving trigonometric equations.