Problem 7 - Precalculus Mathematics for Calculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$\sin x=-\frac{3}{5}, \quad x \text { in Quadrant III }$$

Step-by-Step Solution

Answer

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

Step 1: Identify Trigonometric Range

Since the angle \(x\) is in Quadrant III, both sine and cosine are negative, but tangent is positive. This will guide us in determining the signs of the trigonometric functions.

Step 2: Find \(\cos x\) using Pythagorean Identity

We use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to find \(\cos x\). Substituting \(\sin x = -\frac{3}{5}\), we have \((-\frac{3}{5})^2 + \cos^2 x = 1\). Simplifying, \(\frac{9}{25} + \cos^2 x = 1\). Hence, \(\cos^2 x = 1 - \frac{9}{25} = \frac{16}{25}\), so \(\cos x = -\frac{4}{5}\) (negative, because \(x\) is in Quadrant III).

Step 3: Calculate \(\sin 2x\) using the Double Angle Formula

The double angle formula for sine is \(\sin 2x = 2 \sin x \cos x\). Substituting the values of \(\sin x = -\frac{3}{5}\) and \(\cos x = -\frac{4}{5}\), we calculate \(\sin 2x = 2 \times (-\frac{3}{5}) \times (-\frac{4}{5}) = \frac{24}{25}\).

Step 4: Calculate \(\cos 2x\) using the Double Angle Formula

The double angle formula for cosine is \(\cos 2x = \cos^2 x - \sin^2 x\). Using \(\cos x = -\frac{4}{5}\) and \(\sin x = -\frac{3}{5}\), we have \(\cos 2x = (\frac{16}{25}) - (\frac{9}{25}) = \frac{7}{25}\).

Step 5: Calculate \(\tan 2x\) using the Double Angle Formula

The double angle formula for tangent is \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\). First we find \(\tan x = \frac{\sin x}{\cos x} = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4}\). Then, substituting into the formula, \(\tan 2x = \frac{2 \times \frac{3}{4}}{1 - (\frac{3}{4})^2} = \frac{\frac{3}{2}}{\frac{7}{16}} = \frac{24}{7}\).

Key Concepts

Double Angle Formulas

In trigonometry, double angle formulas are essential for simplifying expressions and solving equations involving angles twice a specific angle. These formulas allow us to express trigonometric functions of double angles, such as \(2x\), in terms of the trigonometric functions of \(x\). This can make complex problems more manageable.

The double angle formulas are as follows:

• \( \sin 2x = 2 \sin x \cos x \)
• \( \cos 2x = \cos^2 x - \sin^2 x \) or \( \cos 2x = 1 - 2\sin^2 x \) or \( \cos 2x = 2\cos^2 x - 1 \)
• \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \)

These formulas are derived using basic trigonometric identities.

When using them, it is crucial to keep in mind the signs of the trigonometric values in their respective quadrants to ensure the correct result, as we saw in the problem. The double angle formulas provide efficient strategies to work through problems encountering the double of an angle.

Trigonometric Identities

Trigonometric identities are essential tools used to simplify trigonometric expressions and solve trigonometric equations. They express a relationship among the trigonometric functions that are true for all values of the included angles. These identities serve as the foundation for solving various trigonometric problems.

Key trigonometric identities include:

• Pythagorean identities: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal identities: \( \csc x = \frac{1}{\sin x} \), \( \sec x = \frac{1}{\cos x} \), \( \cot x = \frac{1}{\tan x} \)
• Quotient identities: \( \tan x = \frac{\sin x}{\cos x} \), \( \cot x = \frac{\cos x}{\sin x} \)

In our given problem, we used the Pythagorean identity to find \(\cos x\) given \(\sin x\), ensuring we respected the sign of \(\cos x\) in Quadrant III.

Understanding these identities helps in changing variables, proving equivalences, and simplifying complex expressions. They are indispensable in calculus and physics where trigonometry is frequently applied.

Quadrants in Trigonometry

The unit circle is divided into four quadrants, and the location of an angle within these quadrants impacts the sign of its trigonometric functions. This feature is integral to solving and understanding trigonometric equations.

In trigonometry:

• Quadrant I: \(\sin\), \(\cos\), \(\tan\) are all positive.
• Quadrant II: \(\sin\) is positive; \(\cos\), \(\tan\) are negative.
• Quadrant III: \(\tan\) is positive; \(\sin\), \(\cos\) are negative.
• Quadrant IV: \(\cos\) is positive; \(\sin\), \(\tan\) are negative.

In the problem, angle \(x\) is in Quadrant III, meaning both \(\sin x\) and \(\cos x\) are negative, and this affects the signs we assign to the functions derived using the double angle formulas.

Understanding the behavior of trigonometric functions in different quadrants is key to ensuring the accuracy of calculations and the correctness of solutions in trigonometry. This knowledge enables us to predict how trigonometric values change with angles and their position on the unit circle.