Problem 7 - Algebra 2 and Trigonometry

Question

In \(3-8,\) for each value of \(\theta,\) use double-angle formulas to find a. \(\sin 2 \theta,\) b. \(\cos 2 \theta,\) c. \(\tan 2 \theta .\) Show all work. $$ \theta=\frac{7 \pi}{6} $$

Step-by-Step Solution

Answer

\(\sin 2\theta = \frac{\sqrt{3}}{2}\), \(\cos 2\theta = \frac{1}{2}\), \(\tan 2\theta = \sqrt{3}\).

Step 1: Determine the formulas for double angles

To solve this exercise, we need to use the double-angle formulas:1. \( \sin 2\theta = 2 \sin \theta \cos \theta \)2. \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) (or equivalent forms)3. \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \)

Step 2: Calculate \(\sin \theta\) and \(\cos \theta\)

Since \(\theta = \frac{7\pi}{6}\), we first need to calculate \(\sin \theta\) and \(\cos \theta\). The angle \(\frac{7\pi}{6}\) is in the third quadrant, where sine and cosine are negative. \[\sin \left(\frac{7\pi}{6}\right) = -\frac{1}{2}\] \[\cos \left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}\]

Step 3: Apply the double-angle formula for \(\sin 2\theta\)

Using the formula \(\sin 2\theta = 2 \sin \theta \cos \theta\), substitute the calculated values for \(\sin \theta\) and \(\cos \theta\):\[\sin 2 \left(\frac{7\pi}{6}\right) = 2 \left(-\frac{1}{2}\right) \left(-\frac{\sqrt{3}}{2}\right)\]\[= \frac{\sqrt{3}}{2}\]

Step 4: Apply the double-angle formula for \(\cos 2\theta\)

Using the formula \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\), substitute in our values:\[\cos 2 \left(\frac{7\pi}{6}\right) = \left(-\frac{\sqrt{3}}{2}\right)^2 - \left(-\frac{1}{2}\right)^2\]\[= \frac{3}{4} - \frac{1}{4}\]\[= \frac{1}{2}\]

Step 5: Use the formula to find \(\tan 2\theta\)

Use the formula \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\). First, find \(\tan \theta\): \[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\].Then substitute into the formula for \(\tan 2\theta\):\[\tan 2 \left(\frac{7\pi}{6}\right) = \frac{2 \left(\frac{1}{\sqrt{3}}\right)}{1 - \left(\frac{1}{\sqrt{3}}\right)^2}\]\[= \frac{\frac{2}{\sqrt{3}}}{1 - \frac{1}{3}}\]\[= \frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}}\]\[= \frac{3}{\sqrt{3}} = \sqrt{3}\]

Step 6: Conclusion: Present the results

The double-angle values for \( \theta = \frac{7\pi}{6} \) are: 1. \( \sin 2\theta = \frac{\sqrt{3}}{2} \)2. \( \cos 2\theta = \frac{1}{2} \)3. \( \tan 2\theta = \sqrt{3} \)

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in solving trigonometry problems efficiently. They are equations involving trigonometric functions that are true for every value of the variable for which the identities are defined. Some commonly used trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Reciprocal Identities: \( \csc \theta = \frac{1}{\sin \theta}, \sec \theta = \frac{1}{\cos \theta}, \cot \theta = \frac{1}{\tan \theta} \)
• Double-Angle Formulas (which we'll explore more later): \( \sin 2\theta, \cos 2\theta, \tan 2\theta \)

Understanding these identities is crucial for simplifying expressions and solving problems that involve angles and their trigonometric functions. They provide a foundation upon which more complex trigonometric problem solving is built.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that describe relationships in a right triangle between an angle and the ratios of two sides.

• Sine Function: For an angle \( \theta \), \( \sin \theta \) is the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine Function: For an angle \( \theta \), \( \cos \theta \) is the ratio of the adjacent side to the hypotenuse in a right triangle.

These functions are periodic, with a period of \( 2\pi \) radians or 360 degrees. Sine and cosine values range from -1 to 1 as the angle changes, which reflects their roles in wave functions.
For example, in the given exercise, sine and cosine are evaluated at \( \theta = \frac{7\pi}{6} \), yielding negative results since this angle lies in the third quadrant where both functions are negative.

Tangent Function

The tangent function relates to the sine and cosine functions and is defined as the ratio of sine to cosine for a given angle \( \theta \). It is represented as:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Tangent is periodic with a period of \( \pi \) radians or 180 degrees. This means the pattern of the tangent function repeats every \( \pi \) radians. The function's values can range from negative to positive infinity, depending on the angle.

• Undefined Points: Tangent is undefined wherever cosine is zero, as division by zero is not possible. In a full cycle, these occur at angles \( \theta = \frac{\pi}{2} + k\pi \).

In our example, calculating \( \tan \theta \) for \( \theta = \frac{7\pi}{6} \) gives \( \frac{1}{\sqrt{3}} \), since both sine and cosine are negative and their ratio is positive.

Double-Angle Identities

Double-angle identities are specific trigonometric identities used to express trigonometric functions of double angles in terms of the function of the original angle. These are particularly helpful in simplifying expressions or solving trigonometric equations.

• Sine Double-Angle Identity: \( \sin 2\theta = 2 \sin \theta \cos \theta \)
• Cosine Double-Angle Identity: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)
• Tangent Double-Angle Identity: \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \)

These formulas allow expressing the trigonometric functions of double angles in terms of known values, which simplifies calculations without a calculator. For \( \theta = \frac{7\pi}{6} \), these identities were used to determine \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \).

Trigonometry Problem Solving

Solving trigonometry problems often involves applying identities to express unknown angles in terms of known values. The goal is to simplify and find exact values for trigonometric functions, especially when they are in standard angles forms.
Some strategies to effectively tackle trigonometry problems include:

• Identifying Given Values and Quadrants: Locate angles on the unit circle and determine their sine and cosine based on their quadrant.
• Applying Known Identities: Use trigonometric identities like the double-angle identities to simplify expressions.
• Practice: Regular practice leads to recognizing patterns and solving similar problems quickly and efficiently.

In the exercise, knowing \( \sin \) and \( \cos \) for \( \theta \) in the third quadrant and applying relevant identities helped find \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \) efficiently.