Problem 4 - Precalculus Mathematics for Calculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$\tan x=-\frac{4}{3}, \quad x \text { in Quadrant II }$$

Step-by-Step Solution

Answer

\(\sin 2x = -\frac{24}{25}\), \(\cos 2x = -\frac{7}{25}\), \(\tan 2x = \frac{24}{7}\).

Step 1: Use Identity for Double Angle Formulas

To find \( \sin 2x \), \( \cos 2x \), and \( \tan 2x \), we will use the double angle formulas:\[\sin 2x = 2 \sin x \cos x, \]\[\cos 2x = \cos^2 x - \sin^2 x,\]\[\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}.\]We'll use these formulas once we know \( \sin x \) and \( \cos x \).

Step 2: Determine the Value of \(\cos x\) and \(\sin x\)

Since \( \tan x = \frac{\sin x}{\cos x} = -\frac{4}{3} \) and \( x \) is in Quadrant II, \( \cos x < 0 \) and \( \sin x > 0 \).We can set \( \cos x = -3k \) and \( \sin x = 4k \) for some \( k \).Using the Pythagorean identity, \( \sin^2 x + \cos^2 x = 1 \), we get:\[(4k)^2 + (-3k)^2 = 1 \16k^2 + 9k^2 = 1 \25k^2 = 1 \k^2 = \frac{1}{25} \k = \frac{1}{5}.\]Thus, \( \sin x = \frac{4}{5} \) and \( \cos x = -\frac{3}{5} \).

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

Use the double angle formula, \( \sin 2x = 2 \sin x \cos x \):\[\sin 2x = 2 \left(\frac{4}{5}\right) \left(-\frac{3}{5}\right) = 2 \left(-\frac{12}{25}\right) = -\frac{24}{25}.\]

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

Use the double angle formula, \( \cos 2x = \cos^2 x - \sin^2 x \):\[\cos 2x = \left(-\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25}.\]

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

Use the double angle formula, \( \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \):\[\tan 2x = \frac{2 \left(-\frac{4}{3}\right)}{1 - \left(-\frac{4}{3}\right)^2} = \frac{-\frac{8}{3}}{1 - \frac{16}{9}} = \frac{-\frac{8}{3}}{-\frac{7}{9}} = \frac{8}{3} \times \frac{9}{7} = \frac{24}{7}.\]

Key Concepts

Trigonometric Identities

Trigonometric identities are foundational in understanding relationships between angles and sides of triangles, particularly right triangles. These identities, such as Pythagorean identities and angle sum formulas, are crucial tools in solving trigonometric equations.
For the problem of finding double angles like \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\), the double angle formulas come in handy:

• \(\sin 2x = 2 \sin x \cos x\)
• \(\cos 2x = \cos^2 x - \sin^2 x\)
• \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\)

These formulas allow us to express trigonometric functions of double angles in terms of the functions of single angles. This simplification is particularly useful when dealing with angles expressed in terms of known trigonometric ratios from the unit circle.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It's a vital component for understanding trigonometric functions because it provides a geometric representation of them. On the unit circle:

• The x-coordinate of a point represents \(\cos \theta\)
• The y-coordinate represents \(\sin \theta\)
• \(\tan \theta\) is the ratio of the y-coordinate to the x-coordinate, or \(\frac{\sin \theta}{\cos \theta}\)

To solve our exercise, the fact that \(\tan x = -\frac{4}{3}\) implies that the points representing \(x\) on the unit circle will have their sine and cosine values in such a ratio, with a negative cosine because \(x\) is in Quadrant II. This information helps determine the exact sine and cosine values for doubled angles using the identities above.

Quadrants in Trigonometry

Understanding quadrants in trigonometry is key to knowing the signs of trigonometric values. The coordinate plane is divided into four quadrants:

• Quadrant I: \(\sin > 0\), \(\cos > 0\)
• Quadrant II: \(\sin > 0\), \(\cos < 0\)
• Quadrant III: \(\sin < 0\), \(\cos < 0\)
• Quadrant IV: \(\sin < 0\), \(\cos > 0\)

Our given \(x\) value resides in Quadrant II, where sine is positive and cosine is negative. This affects not only the computation of \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) using double angle formulas, but also the expected sign of these values in computation. Thus, knowing the quadrant helps anticipate correct results by ensuring the sign of results aligns with the quadrant values.