Problem 8 - Precalculus Mathematics for Calculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\sec x=2, \quad x\) in Quadrant IV

Step-by-Step Solution

Answer

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

Step 1: Understand Trigonometric Identities

To find \( \sin 2x \), \( \cos 2x \), and \( \tan 2x \), we use the double-angle formulas:\[\sin 2x = 2 \sin x \cos x, \quad \cos 2x = \cos^2 x - \sin^2 x, \quad \tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\]We need to find \( \sin x \), \( \cos x \), and \( \tan x \) first.

Step 2: Find Cosine from Secant

Since \( \sec x = 2 \), we have \( \cos x = \frac{1}{\sec x} = \frac{1}{2} \). Since \( x \) is in Quadrant IV, \( \cos x \) is positive in this quadrant.

Step 3: Determine Sine Using Pythagorean Identity

Using the identity \( \sin^2 x + \cos^2 x = 1 \), substitute in \( \cos x = \frac{1}{2} \):\[\sin^2 x + \left(\frac{1}{2}\right)^2 = 1\]\[\sin^2 x + \frac{1}{4} = 1\]\[\sin^2 x = \frac{3}{4}\]\[\sin x = -\frac{\sqrt{3}}{2}\]Since \( x \) is in the fourth quadrant, \( \sin x \) is negative.

Step 4: Calculate Sine of Double Angle

Using \( \sin 2x = 2 \sin x \cos x \), substitute \( \sin x = -\frac{\sqrt{3}}{2} \) and \( \cos x = \frac{1}{2} \):\[\sin 2x = 2 \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{1}{2}\right)\]\[\sin 2x = -\frac{\sqrt{3}}{2}\]

Step 5: Calculate Cosine of Double Angle

Using \( \cos 2x = \cos^2 x - \sin^2 x \), substitute \( \cos x = \frac{1}{2} \) and \( \sin x = -\frac{\sqrt{3}}{2} \):\[\cos 2x = \left(\frac{1}{2}\right)^2 - \left(-\frac{\sqrt{3}}{2}\right)^2\]\[\cos 2x = \frac{1}{4} - \frac{3}{4} = -\frac{1}{2}\]

Step 6: Calculate Tangent of Double Angle

Using \( \tan x = \frac{\sin x}{\cos x} = \frac{-\sqrt{3}/2}{1/2} = -\sqrt{3} \), substitute into \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \):\[\tan 2x = \frac{2(-\sqrt{3})}{1 - (-\sqrt{3})^2}\]\[\tan 2x = \frac{-2\sqrt{3}}{1 - 3} = \frac{-2\sqrt{3}}{-2} = \sqrt{3}\]

Key Concepts

Double Angle Formulas

Double Angle Formulas are a key tool in trigonometry for finding trigonometric values of double angles. These formulas are particularly useful because they express trigonometric functions of \(2x\) in terms of \(x\). This changes a more complex computation into something manageable by leveraging known values of basic trigonometric functions.

• For sine: \(\sin 2x = 2 \sin x \cos x\) represents the sine of double an angle as twice the product of the sine and cosine of the original angle.
• For cosine: \(\cos 2x = \cos^2 x - \sin^2 x\) can also be expressed using variations such as \(1 - 2\sin^2 x\) or \(2\cos^2 x - 1\), making it flexible based on what information is given.
• For tangent: \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\), which involves fractions, ties the double-angle tangent to the single angle tangent.

By applying these formulas, we can calculate the sine, cosine, and tangent of any doubled angle just with the knowledge of the original angle's trigonometric functions. Understanding these formulas is essential in simplifying trigonometric problems, especially when given limited information about the angle.

Pythagorean Identity

The Pythagorean Identity is fundamental in trigonometry. It's based on the Pythagorean theorem and relates the squares of the sine and cosine of an angle. The identity is expressed as \(\sin^2 x + \cos^2 x = 1\). This is pivotal for transforming one trigonometric function in terms of another.
In the context of the given problem, knowing \(\cos x = \frac{1}{2}\), and the identity \(\sin^2 x + \frac{1}{4} = 1\) allowed us to solve for \(\sin x\).

• Subtracting \(\frac{1}{4}\) from both sides gives \(\sin^2 x = \frac{3}{4}\).
• Taking the square root, and considering the quadrant of the angle, we determine that \(\sin x = -\frac{\sqrt{3}}{2}\), since sine is negative in the fourth quadrant.

This identity is not just a tool for finding specific function values but can help in solving complex trigonometric equations and making sense of different function relationships in both mathematical and real-world problem-solving scenarios.

Trigonometric Quadrants

Understanding Trigonometric Quadrants is essential for determining the signs of trigonometric functions based on the angle's position on the coordinate plane.
The circle is divided into four quadrants:

• Quadrant I: All trigonometric functions (sine, cosine, tangent) are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

In our problem, \(x\) is in the fourth quadrant. It means cosine is positive, affirming \(\cos x = \frac{1}{2}\), while sine and tangent are negative, which explains why \(\sin x = -\frac{\sqrt{3}}{2}\) and helps in understanding \(\tan x\) as \(-\sqrt{3}\). This quadrant information is crucial for solving trigonometric equations as it dictates the sign of the function values, leading to the correct results in analyses and calculations.