Problem 7 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\sin 2x = -\frac{3}{5}\), \(\cos 2x = \frac{4}{5}\), \(\tan 2x = -\frac{3}{4}\)

Step 1: Determine the Quadrant

Since \( \tan x = -\frac{1}{3} \), \( x \) is an angle in either the second or the fourth quadrant where tangent is negative. Given that \( \cos x > 0 \), \( x \) is in the fourth quadrant.

Step 2: Use Pythagorean Identity for Cosine

In the fourth quadrant, \( \cos x \) is positive. To find \( \cos x \), start by using the identity \( \tan x = \frac{\sin x}{\cos x} = -\frac{1}{3} \), or \( \sin x = -\frac{1}{3} \cos x\). Using the Pythagorean identity, \( \sin^2 x + \cos^2 x = 1 \), substitute \( \sin x = -\frac{1}{3} \cos x \), leading to:\[ \left( -\frac{1}{3} \cos x \right)^2 + \cos^2 x = 1 \]\[ \frac{1}{9} \cos^2 x + \cos^2 x = 1 \]Combine the terms:\[ \frac{10}{9} \cos^2 x = 1 \]Solve for \( \cos^2 x \):\[ \cos^2 x = \frac{9}{10} \]And hence, \( \cos x = \sqrt{\frac{9}{10}} = \frac{3}{\sqrt{10}} \), because \( x \) is in the fourth quadrant where cosine is positive.

Step 3: Calculate Sine

Given \( \cos x = \frac{3}{\sqrt{10}} \), use \( \tan x = \frac{\sin x}{\cos x} = -\frac{1}{3} \):\[ \sin x = -\frac{1}{3} \cos x = -\frac{1}{3} \cdot \frac{3}{\sqrt{10}} \]Compute \( \sin x \):\[ \sin x = -\frac{1}{\sqrt{10}} \]

Step 4: Use Double Angle Identities

Use the double angle formulas:- \( \sin 2x = 2 \sin x \cos x \)- \( \cos 2x = \cos^2 x - \sin^2 x \)- \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \)

Step 5: Calculate \(\sin 2x\)

Substitute \( \sin x \) and \( \cos x \) into \( \sin 2x = 2 \sin x \cos x \):\[ \sin 2x = 2 \left(-\frac{1}{\sqrt{10}}\right) \left(\frac{3}{\sqrt{10}}\right) \]\[ \sin 2x = 2 \cdot -\frac{3}{10} = -\frac{6}{10} = -\frac{3}{5} \]

Step 6: Calculate \(\cos 2x\)

Substitute \( \cos x = \frac{3}{\sqrt{10}} \) and \( \sin x = -\frac{1}{\sqrt{10}} \) into \( \cos 2x = \cos^2 x - \sin^2 x \):\[ \cos 2x = \left(\frac{3}{\sqrt{10}}\right)^2 - \left(-\frac{1}{\sqrt{10}}\right)^2 \]\[ \cos 2x = \frac{9}{10} - \frac{1}{10} = \frac{8}{10} = \frac{4}{5} \]

Step 7: Calculate \(\tan 2x\)

Use \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \) with \( \tan x = -\frac{1}{3} \):\[ \tan 2x = \frac{2 \cdot -\frac{1}{3}}{1 - \left(-\frac{1}{3}\right)^2} \]\[ \tan 2x = \frac{-\frac{2}{3}}{1 - \frac{1}{9}} = \frac{-\frac{2}{3}}{\frac{8}{9}} \]Simplify:\[ \tan 2x = -\frac{2}{3} \times \frac{9}{8} = -\frac{3}{4} \]

Key Concepts

Double Angle Formulas

Double angle formulas are powerful tools in trigonometry that allow us to express trigonometric functions of double angles in terms of functions of single angles. In the context of the problem, they are instrumental in finding the values of \( \sin 2x \), \( \cos 2x \), and \( \tan 2x \).

Here is a brief rundown of these formulas:

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

These formulas work by leveraging the relationships between the sine, cosine, and tangent of an angle \( x \) to determine these functions at an angle \( 2x \).

In our problem, after finding \( \sin x \) and \( \cos x \), the double angle formulas help us calculate \( \sin 2x = -\frac{3}{5} \), \( \cos 2x = \frac{4}{5} \), and \( \tan 2x = -\frac{3}{4} \). This demonstrates how understanding these identities allows us to solve problems involving trigonometric functions efficiently.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that connects the square of the sine and cosine of an angle to one. Mathematically, it's expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is derived from the Pythagorean theorem applied to the unit circle. It's incredibly useful because it allows us to find one trigonometric function if we know the other.

In this exercise, we use the Pythagorean identity to solve for \( \cos x \) when we already know that \( \tan x = -\frac{1}{3} \). With the transformation \( \tan x = \frac{\sin x}{\cos x} \) and substituting \( \sin x = -\frac{1}{3} \cos x \), we replace \( \sin x \) in the Pythagorean identity with the expression involving \( \cos x \).

Solving the resulting equation helps us find \( \cos x = \frac{3}{\sqrt{10}} \), showing how the identity can aid in determining missing trigonometric values with given information.

Trigonometric Functions

Trigonometric functions are foundational elements in trigonometry, representing the relationships between the angles and sides of right triangles. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each function has its unique properties and plays a crucial role in various mathematical computations.

In the context of this problem:

• \( \sin x \) represents the ratio of the opposite side to the hypotenuse of a right triangle.
• \( \cos x \) represents the adjacent side to the hypotenuse ratio.
• \( \tan x \) is the ratio of the opposite side to the adjacent side.

Understanding these ratios helps solve problems involving any angle, not just those within the realm of right triangles.

The problem gives us \( \tan x = -\frac{1}{3} \), indicating the relationship between \( \sin x \) and \( \cos x \). By appreciating these connections and utilizing identities like those for double angles and the Pythagorean identity, we can explore the values of these functions even at other angles like \( 2x \). Understanding these fundamental trigonometric functions is key to solving complex trigonometric equations.