Problem 9 - Algebra and Trigonometry

Question

\(3-10\) 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: Use the tangent identity to find sine and cosine

We know that \( \tan x = \frac{\sin x}{\cos x} = -\frac{1}{3} \). Since \( \cos x > 0 \), its positive, and \( \sin x \) must be negative because \( \tan x \) is negative. Assume \( \sin x = -k \) and \( \cos x = 3k \). Then \( \tan x = \frac{-k}{3k} = -\frac{1}{3} \). We use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). Replacing, we have \((3k)^2 + (-k)^2 = 1 \), leading to \( 10k^2 = 1 \). Thus, \( k^2 = \frac{1}{10} \) and hence \( k = \frac{1}{\sqrt{10}} \). Therefore, \( \sin x = -\frac{1}{\sqrt{10}} \) and \( \cos x = \frac{3}{\sqrt{10}} \).

Step 2: Calculate \(\sin 2x \) using the double angle formula

The double angle formula for sine is \( \sin 2x = 2 \sin x \cos x \). Using the values obtained for \( \sin x \) and \( \cos x \), substitute: \( \sin 2x = 2 \left(-\frac{1}{\sqrt{10}}\right) \left(\frac{3}{\sqrt{10}}\right) = -\frac{6}{10} = -\frac{3}{5} \).

Step 3: Calculate \(\cos 2x \) using the double angle formula

The double angle formula for cosine is \( \cos 2x = \cos^2 x - \sin^2 x \). With \( \cos x = \frac{3}{\sqrt{10}} \) and \( \sin x = -\frac{1}{\sqrt{10}} \), we have \( \cos^2 x = \frac{9}{10} \) and \( \sin^2 x = \frac{1}{10} \). Thus, \( \cos 2x = \frac{9}{10} - \frac{1}{10} = \frac{8}{10} = \frac{4}{5} \).

Step 4: Calculate \(\tan 2x \) using the double angle formula

The formula for the tangent of a double angle is \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \). Given \( \tan x = -\frac{1}{3} \), we substitute: \( \tan 2x = \frac{2(-\frac{1}{3})}{1 - (-\frac{1}{3})^2} \). Simplify this to get \( \tan 2x = \frac{-\frac{2}{3}}{1 - \frac{1}{9}} = \frac{-\frac{2}{3}}{\frac{8}{9}} = \frac{-2}{3} \cdot \frac{9}{8} = -\frac{3}{4} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations that relate the angles and sides of a triangle. They are essential tools in trigonometry to simplify expressions and solve trigonometric equations. One of the most famous sets of identities includes:

• Pythagorean identities
• Angle sum and difference identities
• Double angle and half angle identities

In the given exercise, the double angle identities are used to find the values of \(\sin 2x, \, \cos 2x, \text{ and } \tan 2x\). Understanding and applying these identities can help solve complex problems involving angles.

Tangent Identity

The tangent identity relates the sine and cosine of an angle, defined as \(\tan x = \frac{\sin x}{\cos x}\). It helps in transforming problems involving tangent into more manageable expressions involving sine and cosine. For instance, in the given exercise, since \(\tan x = -\frac{1}{3}\), the sine and cosine values can be directly derived using this identity. A negative tangent value implies that sine and cosine have opposite signs. Hence, understanding this identity is crucial in deducing unknown trigonometric values.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental trigonometric identities. It is expressed as \(\sin^2 x + \cos^2 x = 1\). This identity is based on the Pythagorean theorem and is valid for any angle x. It allows us to find one trigonometric function if the other is known.
In this exercise, assuming \(\sin x = -k\) and \(\cos x = 3k\) due to \(\tan x = -\frac{1}{3}\), substituting into the identity gives us an equation that we can solve for \(k\). We end up finding that \(k = \frac{1}{\sqrt{10}}\), and therefore the precise values of sine and cosine can be determined. The Pythagorean identity is indispensable in verifying and calculating these values without assuming any particular angles.

Sine and Cosine Values

Sine and cosine values are the building blocks of trigonometry. They represent the ratio of side lengths in a right triangle. In relation to angles, they describe the projections of a point on a circle onto the x and y axes. Their values are crucial in calculating other trigonometric functions.
For this exercise, taking \(\tan x = -\frac{1}{3}\) and knowing \(\cos x > 0\), we use these values in the double angle formulas to find \(\sin 2x\) and \(\cos 2x\). Specifically:

• \(\sin x = -\frac{1}{\sqrt{10}}\)
• \(\cos x = \frac{3}{\sqrt{10}}\)

These exact values are crucial as they are inserted into the double angle formulas. Understanding how to find or use these trigonometric values is essential for tackling many problems in trigonometry effectively.