Problem 29 - Precalculus with Calculus Previews

Question

Use the given information to find (a) \(\cos 2 x,(b) \sin 2 x\), and \((c) \tan 2 x\). $$ \sin x=\sqrt{2} / 3, \quad \pi / 2

Step-by-Step Solution

Answer

(a) \(\frac{5}{9}\), (b) \(-\frac{2\sqrt{14}}{9}\), (c) \(-\frac{2\sqrt{14}}{5}\).

Step 1: Use the Identity for \(\cos x\)

Given \(\sin x = \frac{\sqrt{2}}{3}\), use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to find \(\cos x\). First, calculate \(\sin^2 x\):\[\sin^2 x = \left(\frac{\sqrt{2}}{3}\right)^2 = \frac{2}{9}\]Then, substitute into the identity:\[\cos^2 x = 1 - \sin^2 x = 1 - \frac{2}{9} = \frac{7}{9}\]Since \(\pi/2 < x < \pi\), \(\cos x\) is negative, so \(\cos x = -\sqrt{\frac{7}{9}} = -\frac{\sqrt{7}}{3}\).

Step 2: Calculate \(\cos 2x\) Using the Double Angle Formula

The double angle formula for cosine is \(\cos 2x = 2 \cos^2 x - 1\). Substitute the value of \(\cos^2 x\):\[\cos 2x = 2 \left(-\frac{\sqrt{7}}{3}\right)^2 - 1\]\[= 2 \times \frac{7}{9} - 1 = \frac{14}{9} - 1 = \frac{14}{9} - \frac{9}{9} = \frac{5}{9}\]

Step 3: Calculate \(\sin 2x\) Using the Double Angle Formula

The double angle formula for sine is \(\sin 2x = 2 \sin x \cos x\). Substitute the known values:\[\sin 2x = 2 \times \frac{\sqrt{2}}{3} \times \left(-\frac{\sqrt{7}}{3}\right)\]\[= -2 \times \frac{\sqrt{14}}{9} = -\frac{2\sqrt{14}}{9}\]

Step 4: Calculate \(\tan 2x\) Using the Identity \(\tan 2x = \frac{\sin 2x}{\cos 2x}\)

Substitute \(\sin 2x\) and \(\cos 2x\) from the previous steps:\[\tan 2x = \frac{-\frac{2\sqrt{14}}{9}}{\frac{5}{9}}\]\[= -\frac{2\sqrt{14}}{5}\]

Step 5: Verify the Signs for Each Trigonometric Function

Verify that the calculated trigonometric values align with expectations given the quadrant information. Since \(\pi < 2x < 2\pi\), \(\cos 2x\) can be positive, \(\sin 2x\) is negative, and \(\tan 2x\) is negative, consistent with our results.

Key Concepts

Double Angle Formulas

Double Angle Formulas are useful in trigonometry for calculating functions like sine, cosine, and tangent at twice the angle of a given angle. They can simplify complex expressions involving trigonometric functions. The three main double angle formulas are:

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

For problems where you know \( \sin x \) and \( \cos x \), these formulas give you a way to determine trigonometric values for \( 2x \). Knowing these shortcuts can save you a lot of work and simplify your calculations.
In the context of the given problem, we use the double angle formulas to find \( \cos 2x\), \( \sin 2x\), and \( \tan 2x\) by first calculating \( \cos x \) using initial data and then applying these formulas step by step. It’s important to also regard the quadrant for signs, like in our original exercise where each value's sign was checked to match the quadrant.

Pythagorean Identity

The Pythagorean Identity is fundamental in trigonometry and often used for transforming one function into another. The most familiar form is:\[\sin^2 x + \cos^2 x = 1\]This identity is very useful, particularly when one of the trigonometric values is known, and you need to find another.

• If you know \( \sin x \), you can compute \( \cos x \) using: \( \cos^2 x = 1 - \sin^2 x \)
• Similarly, if \( \cos x \) is known, \( \sin x \) can be found with: \( \sin^2 x = 1 - \cos^2 x \)

In our example, given \( \sin x = \frac{\sqrt{2}}{3} \), the Pythagorean Identity was used to figure out \( \cos x \).
We squared \( \sin x \), substituted \( \sin^2 x \) in the identity, and solved for \( \cos x \). Knowing the quadrant helps determine the sign of \( \cos x \). This identity is a classic tool simplifying how to work with trigonometric problems, as it links sine and cosine straightforwardly.

Trigonometric Functions

Trigonometric Functions are the building blocks of trigonometry, essential in understanding angles and cycles. These functions include sine, cosine, and tangent, with definitions rooted in right triangle geometry and the unit circle.

• \( \sin x \), or sine, is the ratio of the length of the side opposite the angle to the hypotenuse.
• \( \cos x \), or cosine, represents the ratio of the length of the adjacent side to the hypotenuse.
• \( \tan x \), or tangent, is the ratio of the opposite side to the adjacent side, which can also be expressed as \( \frac{\sin x}{\cos x} \).

Understanding these basic functions is key, as they are used across a multitude of mathematical and real-world applications like physics, engineering, and graphical modeling.
For our given problem, the calculations found \( \sin 2x \), \( \cos 2x \), and \( \tan 2x \) based on these definitions.
These functions were also checked within the context of the unit circle's quadrants to ensure the right sign as per their cyclical nature.