Problem 32 - Precalculus with Calculus Previews

Question

Use the given information to find (a) \(\cos 2 x,(b) \sin 2 x\), and \((c) \tan 2 x\). $$ \csc x=-3, \pi

Step-by-Step Solution

Answer

(a) \( \cos 2x = \frac{7}{9} \), (b) \( \sin 2x = \frac{4\sqrt{2}}{9} \), (c) \( \tan 2x = \frac{4\sqrt{2}}{7} \).

Step 1: Identify Known Values

We know that \( \csc x = -3 \), which means that \( \sin x = -\frac{1}{3} \). The interval \( \pi < x < \frac{3\pi}{2} \) indicates that \( x \) lies in the third quadrant, where sine is negative, aligning with our calculation.

Step 2: Calculate \( \cos x \) Using Pythagorean Identity

Use the Pythagorean identity to find \( \cos x \):\[sin^2 x + cos^2 x = 1 \]Substituting \( \sin x \):\[\left(-\frac{1}{3}\right)^2 + \cos^2 x = 1 \\frac{1}{9} + \cos^2 x = 1 \\cos^2 x = 1 - \frac{1}{9} = \frac{8}{9} \\cos x = -\sqrt{\frac{8}{9}} = -\frac{2\sqrt{2}}{3} \\]Since \( x \) is in the third quadrant, \( \cos x \) is negative.

Step 3: Compute \( \cos 2x \) Using Double Angle Formula

The formula for \( \cos 2x \) is:\[\cos 2x = 2\cos^2 x - 1\]Substitute the value for \( \cos x \):\[\cos 2x = 2\left(-\frac{2\sqrt{2}}{3}\right)^2 - 1 \= 2 \times \frac{8}{9} - 1 \= \frac{16}{9} - 1 \= \frac{16}{9} - \frac{9}{9} \= \frac{7}{9} \]

Step 4: Compute \( \sin 2x \) Using Double Angle Formula

The formula for \( \sin 2x \) is:\[\sin 2x = 2\sin x \cos x\]Substitute the values for \( \sin x \) and \( \cos x \):\[\sin 2x = 2 \left(-\frac{1}{3}\right) \left(-\frac{2\sqrt{2}}{3}\right) = 2 \times \frac{2\sqrt{2}}{9} = \frac{4\sqrt{2}}{9}\]

Step 5: Compute \( \tan 2x \) Using the Quotient of Sine and Cosine

The formula for \( \tan 2x \) is:\[\tan 2x = \frac{\sin 2x}{\cos 2x}\]Substitute the values calculated for \( \sin 2x \) and \( \cos 2x \):\[\tan 2x = \frac{\frac{4\sqrt{2}}{9}}{\frac{7}{9}} = \frac{4\sqrt{2}}{7}\]

Key Concepts

Double Angle Formulas

Trigonometric identities are crucial for simplifying and calculating angles. One category of these identities is the double angle formulas. These are formulas that allow us to find the sine, cosine, or tangent of double an angle using known values of the original angle.

Among the double angle formulas, you will most commonly use:

• The cosine double angle formula: \[ \cos 2x = 2\cos^2 x - 1 \]This formula says that you can calculate the cosine of twice an angle by doubling the square of its cosine and subtracting one.
• The sine double angle formula: \[ \sin 2x = 2 \sin x \cos x \]Here, the formula lets you find the sine of double an angle by multiplying two times the sine and cosine of that angle.
• The tangent double angle formula is also important, and is derived as: \[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \]though in practice, we often use \[ \tan 2x = \frac{\sin 2x}{\cos 2x} \]to compute it.

These formulas help simplify the process of finding these trigonometric values, especially when the angle itself is not an easily recognizable one. By mastering these formulas, you will be able to solve a wide variety of mathematical problems more efficiently.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry connecting the sine and cosine of an angle:

• \[ \sin^2 x + \cos^2 x = 1 \]This equation arises from the Pythagorean theorem and is valid for any angle. It confirms that the sum of the squares of the sine and cosine of an angle is always one.

To utilize this identity effectively, consider the context given in a problem. For example, if you are given a sine or cosine value, you can easily solve for the other value using this identity. Let’s use it in practice:

• Given \( \sin x = -\frac{1}{3} \), substitute into the identity: \[ \left( -\frac{1}{3} \right)^2 + \cos^2 x = 1 \] Simplifying gives: \[ \cos^2 x = 1 - \frac{1}{9} \] \[ \cos^2 x = \frac{8}{9} \]And because \( x \) is in the third quadrant, \( \cos x \) is negative: \[ \cos x = -\frac{2\sqrt{2}}{3} \]This helps us compute other trigonometric values using identities such as the double angle formulas.

Quadrant Analysis

Understanding which quadrant an angle lies in helps determine the sign of its trigonometric values. Each quadrant on the unit circle has specific rules for whether the trigonometric functions are positive or negative:

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine is positive, while cosine and tangent are negative.
• Third Quadrant: Tangent is positive, while sine and cosine are negative.
• Fourth Quadrant: Cosine is positive, while sine and tangent are negative.

For example, if you know \( x \) is between \( \pi \) and \( \frac{3\pi}{2} \), it lies in the third quadrant. That means both \( \sin x \) and \( \cos x \) are negative.

This analysis is crucial when determining the signs of trigonometric results. It serves as a reliable guide in solving problems, ensuring you assign the correct sign based on where the angle is located on the unit circle. Knowing which quadrant the angle is in allows you to apply the correct sign to any trigonometric value obtained. Thus, it helps avoid errors in calculations, resulting in more accurate solutions.