Problem 8 - Algebra and Trigonometry

Question

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\cot x=\frac{2}{3}, \quad \sin x>0\)

Step-by-Step Solution

Answer

\( \sin 2x = \frac{12}{13}, \cos 2x = -\frac{5}{13}, \tan 2x = -\frac{12}{5} \).

Step 1: Understand Given Information

We are given that \( \cot x = \frac{2}{3} \) and \( \sin x > 0 \). The condition \( \sin x > 0 \) implies that \( x \) is in Quadrant I or II.

Step 2: Use the Identity for Cotangent and Determine Sine and Cosine

Since \( \cot x = \frac{2}{3} \), then \( \tan x = \frac{3}{2} \). Tangent is positive, which means \( x \) is in Quadrant I where both \( \sin x \) and \( \cos x \) are positive. Using the identity \( \tan x = \frac{\sin x}{\cos x} \), let \( \sin x = 3k \) and \( \cos x = 2k \). Therefore, \( \tan x = \frac{3k}{2k} = \frac{3}{2} \).

Step 3: Determine the Value of k

Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), we substitute \( \sin x = 3k \) and \( \cos x = 2k \): \[ (3k)^2 + (2k)^2 = 1 \]\[ 9k^2 + 4k^2 = 1 \]\[ 13k^2 = 1 \]\[ k^2 = \frac{1}{13} \]\[ k = \frac{1}{\sqrt{13}} \]

Step 4: Calculate Sine and Cosine

Substitute \( k \) back to find \( \sin x \) and \( \cos x \):\[ \sin x = 3k = \frac{3}{\sqrt{13}} \]\[ \cos x = 2k = \frac{2}{\sqrt{13}} \]

Step 5: Find \( \sin 2x \) Using Double Angle Identity

Use the double angle identity \( \sin 2x = 2 \sin x \cos x \):\[ \sin 2x = 2 \times \frac{3}{\sqrt{13}} \times \frac{2}{\sqrt{13}} = 2 \times \frac{6}{13} = \frac{12}{13} \]

Step 6: Find \( \cos 2x \) Using Double Angle Identity

Use the double angle identity \( \cos 2x = \cos^2 x - \sin^2 x \):\[ \cos 2x = \left(\frac{2}{\sqrt{13}}\right)^2 - \left(\frac{3}{\sqrt{13}}\right)^2 \]\[ \cos 2x = \frac{4}{13} - \frac{9}{13} \]\[ \cos 2x = -\frac{5}{13} \]

Step 7: Find \( \tan 2x \) Using Identity

Use the identity \( \tan 2x = \frac{\sin 2x}{\cos 2x} \):\[ \tan 2x = \frac{\frac{12}{13}}{-\frac{5}{13}} = -\frac{12}{5} \]

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding relationships within a right triangle. They are primarily used to express the ratio of sides of the triangle. Here are the main functions:

• Sine (\( ext{sin}\)): This function represents the ratio of the opposite side to the hypotenuse.
• Cosine (\( ext{cos}\)): This is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( ext{tan}\)): This refers to the ratio of the opposite side to the adjacent side.
• Cotangent (\( ext{cot}\)): The reciprocal of tangent; adjacent over opposite.

Understanding these functions through their ratios helps in solving many problems, especially when you know certain angles or sides of a triangle.

Quadrants

When we talk about angles in trigonometry, we often refer to their positions in different quadrants of the coordinate plane. Each quadrant has distinct characteristics for trigonometric functions:

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

In our example, where \( ext{sin}\) is positive and \( ext{cot}\) is given, we're directed toward Quadrant I or II. However, since tangent (\( an x = \frac{3}{2}\)) is positive, it confirms that the angle \(x\) is in Quadrant I.

Double Angle Formulas

Double angle formulas are powerful tools that help simplify many trigonometric expressions, especially when doubling an angle. Here are the critical formulas you need to know:

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

These formulas allow us to find new expressions like \( ext{sin } 2x\), \( ext{cos } 2x\), or \( ext{tan } 2x\) from known values of sin and cos.

Pythagorean Identity

The Pythagorean identity is fundamental in trigonometry, highlighting the intrinsic relationship between sine and cosine. This identity is expressed as:\[\sin^2 x + \cos^2 x = 1\]Using this identity, if you know one of these functions, you can find the other. For example, if you know \( ext{sin} x\), you can rearrange the identity to solve for \( ext{cos} x\):\[\cos^2 x = 1 - \sin^2 x\]In our exercise, this identity was essential to determine the value of \(k\) and thus find specific values for both \( ext{sin} x\) and \( ext{cos} x\), which are crucial for proceeding to the double angle calculations.