Problem 4 - Contemporary Precalculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\cos x=-\frac{1}{3} \quad\left(\frac{\pi}{2}

Step-by-Step Solution

Answer

Given that \(\cos x = -\frac{1}{3}\) and \(\frac{\pi}{2} < x < \pi\), we have calculated the following double angle values: 1. \(\sin 2x = -\frac{4\sqrt{2}}{3}\) 2. \(\cos 2x = -\frac{7}{9}\) 3. \(\tan 2x = \frac{12\sqrt{2}}{7}\)

Step 1: Find \(\sin x\)

We are given \(\cos x = -\frac{1}{3}\) and that \(\frac{\pi}{2} < x < \pi\). Since \(x\) is in the second quadrant, we know that \(\sin x > 0\). We can use the Pythagorean identity to find \(\sin x\): $$\sin^2 x + \cos^2 x = 1$$ $$\sin^2 x = 1 - \cos^2 x = 1 - \left(-\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}$$ So, we have \(\sin x = ±\sqrt{\frac{8}{9}}\). Since \(\sin x\) is positive in the second quadrant, we take the positive square root: \(\sin x = \sqrt{\frac{8}{9}}\).

Step 2: Find \(\sin 2x\)

Using the double angle formula for sine, we have: $$\sin 2x = 2\sin x\cos x = 2\cdot \sqrt{\frac{8}{9}} \cdot \left(-\frac{1}{3}\right) = -\frac{4\sqrt{2}}{3}$$

Step 3: Find \(\cos 2x\)

Using the double angle formula for cosine, we have: $$\cos 2x = \cos^2 x - \sin^2 x = \left(-\frac{1}{3}\right)^2 - \left(\sqrt{\frac{8}{9}}\right)^2 = \frac{1}{9} - \frac{8}{9} = -\frac{7}{9}$$

Step 4: Find \(\tan 2x\)

To find \(\tan 2x\), we can either use the double angle formula for tangent, or divide \(\sin 2x\) by \(\cos 2x\). We will use the latter method: $$\tan 2x = \frac{\sin 2x}{\cos 2x} = \frac{-\frac{4\sqrt{2}}{3}}{-\frac{7}{9}} = \frac{4\sqrt{2}}{3} \cdot \frac{9}{7} = \frac{12\sqrt{2}}{7}$$ In conclusion, we have: $$\sin 2x = -\frac{4\sqrt{2}}{3}$$ $$\cos 2x = -\frac{7}{9}$$ $$\tan 2x = \frac{12\sqrt{2}}{7}$$

Key Concepts

Double Angle Formulas

Double angle formulas are powerful tools in trigonometry, used to express trigonometric functions at double angles. These are especially helpful for solving problems involving angles that are multiples of known angles. The three primary double angle formulas are:

• For sine: \( \sin 2x = 2 \sin x \cos x \)
• For cosine: \( \cos 2x = \cos^2 x - \sin^2 x \)
• For tangent: \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \)

In our exercise, we used these formulas to find \( \sin 2x \), \( \cos 2x \), and \( \tan 2x \) based on the given \( \cos x \). Knowing one trigonometric function of an angle and its quadrant allows you to calculate the rest. This demonstrates the utility of these identities in finding values beyond simple angles.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry that connects the squares of the sine and cosine functions. The identity states:
\[ \sin^2 x + \cos^2 x = 1 \]
This identity is derived from the Pythagorean theorem and applies to any angle. It is often used to find one trigonometric function when the other is known. For instance, in the provided exercise, we found \( \sin x \) using the known value of \( \cos x \) and the Pythagorean identity. Knowing \( \cos x = -\frac{1}{3} \), we rearranged the identity to find
\[ \sin^2 x = 1 - \left( \cos x \right)^2 = 1 - \left( -\frac{1}{3} \right)^2 = \frac{8}{9} \]
This allowed us to determine \( \sin x \) in the second quadrant as \( \sqrt{\frac{8}{9}} \). The Pythagorean identity is pivotal in converting between trigonometric functions and solving trigonometric equations.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are essential in understanding angles and triangles. They describe the relationships between the angles and sides of a right triangle. Here's what you should know about them:

• Sine (\( \sin \)): Opposite side over hypotenuse in a right triangle.
• Cosine (\( \cos \)): Adjacent side over hypotenuse in a right triangle.
• Tangent (\( \tan \)): Opposite side over adjacent side, or \( \sin \) over \( \cos \).

In our task, we worked in the second quadrant, where certain signs of trigonometric functions change:

• \( \sin x \) is positive.
• \( \cos x \) is negative.
• \( \tan x = \frac{\sin x}{\cos x} \) is negative.

Understanding these signs and relationships helps in finding consistent results when solving trigonometric problems. They are used in various calculations, such as those involving double angles as shown in this exercise.