Problem 3 - Precalculus Mathematics for Calculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$\sin x=\frac{5}{13}, \quad x \text { in Quadrant } \mathbf{I}$$

Step-by-Step Solution

Answer

\(\sin 2x = \frac{120}{169}\), \(\cos 2x = \frac{119}{169}\), \(\tan 2x = \frac{120}{119}\).

Step 1: Determine Cosine of x

Given that \(\sin x = \frac{5}{13}\), we first use the Pythagorean identity to find \(\cos x\). The identity is \(\sin^2 x + \cos^2 x = 1\). Substitute the given sine value: \((\frac{5}{13})^2 + \cos^2 x = 1\). Simplifying, we get \(\frac{25}{169} + \cos^2 x = 1\). Thus, \(\cos^2 x = 1 - \frac{25}{169} = \frac{144}{169}\). Therefore, \(\cos x = \frac{12}{13}\) as \(x\) is in Quadrant I, where cosine is positive.

Step 2: Calculate Sin 2x Using Double Angle Formula

To find \(\sin 2x\), use the double angle formula: \(\sin 2x = 2 \sin x \cos x\). Substitute \(\sin x = \frac{5}{13}\) and \(\cos x = \frac{12}{13}\): \(\sin 2x = 2 \times \frac{5}{13} \times \frac{12}{13} = \frac{120}{169}\).

Step 3: Calculate Cos 2x Using Double Angle Formula

To find \(\cos 2x\), we use the double angle formula: \(\cos 2x = \cos^2 x - \sin^2 x\). Substitute \(\cos x = \frac{12}{13}\) and \(\sin x = \frac{5}{13}\): \(\cos 2x = (\frac{12}{13})^2 - (\frac{5}{13})^2 = \frac{144}{169} - \frac{25}{169} = \frac{119}{169}\).

Step 4: Calculate Tan 2x Using Double Angle Formula

To find \(\tan 2x\), use the formula: \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\). First, find \(\tan x = \frac{\sin x}{\cos x} = \frac{5/13}{12/13} = \frac{5}{12}\). Then, \(\tan 2x = \frac{2 \times \frac{5}{12}}{1 - (\frac{5}{12})^2} = \frac{\frac{10}{12}}{1 - \frac{25}{144}}\). Simplify \(\tan 2x = \frac{\frac{5}{6}}{\frac{119}{144}} = \frac{5}{6} \times \frac{144}{119} = \frac{120}{119}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry that help to simplify and solve various equations. They are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined. Often used in mathematical proofs and problem solving, these identities can transform complex expressions into simpler ones.

Some essential trigonometric identities include:

• Pythagorean identities such as \(\sin^2 x + \cos^2 x = 1\)
• Double angle formulas like \(\sin 2x = 2 \sin x \cos x\) and \(\cos 2x = \cos^2 x - \sin^2 x\)
• Reciprocal identities, for instance, \( \sin x = \frac{1}{\csc x} \)

Understanding and using these identities is crucial, especially in scenarios involving angle transformations, like when dealing with double angle problems. They provide the base for solving more advanced trigonometric equations and are frequently used in calculus, physics, and geometry.

Pythagorean Identity

The Pythagorean identity is one of the most important in trigonometry. It establishes a relationship between the sine and cosine of an angle, stemming from the Pythagorean theorem in a right-angled triangle. The identity is expressed as:

• \(\sin^2 x + \cos^2 x = 1\)

This equation indicates that if you know the sine of an angle, you can find the cosine, and vice versa. Since they are squares in the identity, often the positive value is used in specific quadrants where sine or cosine are positive.

For example, if \(\sin x = \frac{5}{13}\), you can find \(\cos x\) by rearranging the identity: \(\cos^2 x = 1 - \sin^2 x\). Then plug in the values to solve for \(\cos x\). In the exercise above, this method was pivotal in determining that \(\cos x = \frac{12}{13}\), detailing the interdependent nature of trigonometric functions.

Quadrant Analysis

Quadrant analysis is a technique used to determine the sign (positive or negative) of trigonometric functions based on the angle's location. Quadrants are the four sections of the Cartesian plane, divided by the x and y axes. Each quadrant possesses unique characteristics:

• 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 the current problem, the angle \(x\) is in Quadrant I, so both \(\sin x\) and \(\cos x\) are positive. This analysis ensures that when solving for cosine using a square root from the Pythagorean identity, the positive root is selected. Proper quadrant analysis avoids potential errors in sign selection and confirms the correct operation of trigonometric identities across different mathematical contexts.

Sine and Cosine Functions

The sine and cosine functions are the two cornerstone functions in trigonometry, describing the properties of a right triangle or the unit circle. The sine function relates the angle to the ratio of the opposite side over the hypotenuse, while cosine does the same for the adjacent side over the hypotenuse.These functions are periodic, repeating every \(2\pi\) radians, and are used to model periodic phenomena like sound waves. They share a unique relationship via the Pythagorean identity, enhancing their use in analyzing angles, waves, and signals.

In the given problem, knowing \(\sin x\), we used the Pythagorean identity to find \(\cos x\). Then both values aided in computing double angle formulas, demonstrating the intricate interplay between sine and cosine in transforming angles. In essence, understanding these functions is key to mastering trigonometry and applying it effectively across varied mathematical problems and practical applications.