Problem 54 - Precalculus Mathematics for Calculus

Question

Evaluate each expression under the given conditions. \(\tan 2 \theta ; \cos \theta=\frac{3}{5}, \theta\) in Quadrant I

Step-by-Step Solution

Answer

\(\tan 2\theta = -\frac{24}{7}\)

Step 1: Identify the Relevant Trigonometric Identity

To find \(\tan 2\theta\), we use the double angle identity: \(\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}\). To apply this identity, we need to find \(\tan \theta\).

Step 2: Use the Given Value of \\(\cos \theta\\)

We know \(\cos \theta = \frac{3}{5}\). Given that \(\theta\) is in the first quadrant, \(\sin \theta\) will be positive. We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin \theta\).

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

Substitute the known value in the identity: \(\sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1\). This means \(\sin^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}\), giving \(\sin \theta = \frac{4}{5}\) since \(\theta\) is in the first quadrant and \(\sin \theta\) is positive.

Step 4: Calculate \\(\tan \theta\\)

Now that we know \(\sin \theta\) and \(\cos \theta\), we find \(\tan \theta\) using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting the known values, \(\tan \theta = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}\).

Step 5: Evaluate \\(\tan 2\theta\\) Using the Identity

Now substitute \(\tan \theta = \frac{4}{3}\) into the double angle identity: \(tan 2\theta = \frac{2 \cdot \frac{4}{3}}{1 - \left(\frac{4}{3}\right)^2}\). This simplifies to \(\tan 2\theta = \frac{\frac{8}{3}}{1 - \frac{16}{9}} = \frac{\frac{8}{3}}{-\frac{7}{9}}\).

Step 6: Simplify the Expression

Simplify the fraction: \(\tan 2\theta = \frac{8}{3} \cdot \frac{9}{-7} = \frac{72}{-21} = -\frac{24}{7}\). Therefore, \(\tan 2\theta = -\frac{24}{7}\).

Key Concepts

Double Angle Formula

The Double Angle Formula is a powerful tool in trigonometry and is specifically useful for calculating the trigonometric values of angles that are multiples of a given angle. When dealing with angles like \(2\theta\), these formulas help simplify complex expressions into more manageable equations.

For tangent, the formula is:

• \( \tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta} \)

Here, the formula allows us to express \( \tan 2\theta \) in terms of \( \tan \theta \), which can be helpful when the individual tangent is known or can be easily calculated.

By using this formula, we transformed the problem of finding \( \tan 2\theta \) into a series of algebraic manipulations that begin with simpler trigonometric values like \( \cos \theta \) and \( \sin \theta \). This method essentially bridges simpler trigonometric functions with their more complex counterparts, making problem-solving more approachable.

Pythagorean Identity

The Pythagorean Identity is one of the foundational identities in trigonometry, stemming from the Pythagorean theorem. It relates the sine and cosine of an angle and provides a crucial link between these primary trigonometric functions.

The identity can be expressed as:

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

This identity is indispensable, particularly when one trigonometric function is known and you need to deduce another. In our exercise, we were given \( \cos \theta = \frac{3}{5} \). Using the Pythagorean Identity, we can find \( \sin \theta \).

Since \( \theta \) is located in the first quadrant where all trigonometric functions are positive, the identity allowed us to compute \( \sin \theta \) and verify that it satisfied all quadrant considerations. The computations unveiled that \( \sin \theta = \frac{4}{5} \). Not only does the Pythagorean Identity help solve equations, but it also provides insight into the inherent relationships within trigonometric functions.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent describe the ratios of the sides of a right triangle relative to an angle, \( \theta \). In our exercise, understanding these functions was key to finding \( \tan 2\theta \).

Let's briefly revisit the main functions:

• Sine (\( \sin \theta \)): The ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)): The ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): The ratio of the opposite side to the adjacent side, or equivalently \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

With \( \cos \theta \) provided, it directly led to \( \tan \theta \) by first finding \( \sin \theta \). Using these principles and relationships, \( \tan \theta \) was calculated as \( \frac{4}{3} \).

These functions are interconnected through trigonometric identities and exist as part of a broader framework used to analyze periodic phenomena, complex rotations, and more. Recognizing their interdependence through identities like the double angle and Pythagorean identity equips students with versatile tools in mathematics.