Problem 6 - Precalculus Mathematics for Calculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$\csc x=4, \quad \tan x<0$$

Step-by-Step Solution

Answer

\( \sin 2x = -\frac{\sqrt{15}}{8}, \cos 2x = \frac{7}{8}, \tan 2x = -\frac{\sqrt{15}}{7} \)

Step 1: Identify Known Values

We are given that \( \csc x = 4 \). Since \( \csc x = \frac{1}{\sin x} \), we know that \( \sin x = \frac{1}{4} \). Furthermore, \( \tan x < 0 \) helps us identify the quadrant in which \( x \) is located.

Step 2: Determine the Cosine of x

Since \( \sin x = \frac{1}{4} \), we use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to find \( \cos x \).\[ \cos^2 x = 1 - \sin^2 x = 1 - \left( \frac{1}{4} \right)^2 = \frac{15}{16} \]Taking the square root, \( \cos x = \pm \frac{\sqrt{15}}{4} \). Given \( \tan x < 0 \), and knowing \( \sin x > 0 \) (since \( \csc x > 0 \)), \( \cos x \) must be negative. Thus, \( \cos x = -\frac{\sqrt{15}}{4} \).

Step 3: Compute \( \sin 2x \)

Utilize the double angle identity for sine:\[ \sin 2x = 2 \sin x \cos x \]Plug in the known values:\[ \sin 2x = 2 \left( \frac{1}{4} \right)\left( -\frac{\sqrt{15}}{4} \right) = -\frac{\sqrt{15}}{8} \]

Step 4: Compute \( \cos 2x \)

Use the double angle identity for cosine:\[ \cos 2x = \cos^2 x - \sin^2 x \]Plug in the previously found values:\[ \cos 2x = \left( -\frac{\sqrt{15}}{4} \right)^2 - \left( \frac{1}{4} \right)^2 = \frac{15}{16} - \frac{1}{16} = \frac{14}{16} = \frac{7}{8} \]

Step 5: Compute \( \tan 2x \)

Use the identity for the double angle of tangent:\[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \]First, find \( \tan x = \frac{\sin x}{\cos x} = \frac{1/4}{-\sqrt{15}/4} = -\frac{1}{\sqrt{15}} \).Now, plug into the formula:\[ \tan 2x = \frac{2 \left( -\frac{1}{\sqrt{15}} \right)}{1 - \left( -\frac{1}{\sqrt{15}} \right)^2} = \frac{-\frac{2}{\sqrt{15}}}{1 - \frac{1}{15}} = \frac{-\frac{2}{\sqrt{15}}}{\frac{14}{15}} = -\frac{2}{\sqrt{15}} \times \frac{15}{14} = -\frac{\sqrt{15}}{7} \]

Key Concepts

Double Angle Formulas

The double angle formulas in trigonometry allow us to express trigonometric functions of double angles, like \(2x\), in terms of the single angle \(x\). These formulas include identities for sine, cosine, and tangent:

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

All these identities help in simplifying complex trigonometric expressions involving double angles.
For example, to find \(\sin 2x\), you simply multiply 2 by the sine of \(x\) and the cosine of \(x\). This provides a streamlined method to calculate the sine of a double angle.
The double angle for cosine involves the difference of the squares of \(\cos x\) and \(\sin x\), illustrating a neat connection between these two trigonometric functions. This identity also highlights how the cosine function relates fundamentally to rotations and angles in trigonometry.
Lastly, the double angle identity for tangent expresses \(\tan 2x\) in terms of the tangent of \(x\) itself, which helps break down and analyze more complicated trigonometric problems.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and describe relationships between the sides and angles of triangles. The primary trigonometric functions include sine \(\sin\), cosine \(\cos\), and tangent \(\tan\). Each function has unique relationships and definitions.

• \(\sin x\) represents the ratio of the opposite side to the hypotenuse in a right triangle.
• \(\cos x\) is the ratio of the adjacent side to the hypotenuse.
• \(\tan x\) is the ratio of the opposite side to the adjacent side, or equivalently \(\frac{\sin x}{\cos x}\).

In practical applications, these functions help in solving problems involving angles and distances. One example involves boats navigating using directional angles or engineers measuring heights and distances.
Sine and cosine functions also form the basis for many transformations and wave functions, making them essential in fields like physics, engineering, and signal processing. Understanding these functions is crucial for mastering more advanced mathematical concepts.
Additionally, knowing how to transform and manipulate these functions with identities like double angles can aid significantly in solving complex trigonometric problems.

Pythagorean Identity

The Pythagorean identity is one of the cornerstone identities in trigonometry, expressing the fundamental relationship between sine and cosine. It is given by:

• \(\sin^2 x + \cos^2 x = 1\)

Essentially, this identity is derived from the Pythagorean theorem in geometry, which relates to the sides of a right triangle.
This identity not only serves as a critical tool for verifying the correctness of trigonometric relationships but also helps in deriving other identities, like the double angle formulas, discussed earlier.
By using the Pythagorean identity, one can calculate an unknown trigonometric function if the other is known. For instance, if you know \(\sin x\), you can find \(\cos x\) using this identity and vice versa.
The Pythagorean identity underscores the intrinsic relationship between the circular motion described by sine and cosine functions, where the sum of their squared values always equals unity. This reflects the unit circle model in trigonometry, tying together the concepts of angles and circular paths seamlessly.