Problem 30 - Precalculus with Calculus Previews

Question

Use the given information to find (a) \(\cos 2 x,(b) \sin 2 x\), and \((c) \tan 2 x\). $$ \cos x=\sqrt{3} / 5, \quad 3 \pi / 2

Step-by-Step Solution

Answer

\( \cos 2x = -\frac{19}{25}, \sin 2x = -\frac{2\sqrt{66}}{25}, \tan 2x = \frac{2\sqrt{66}}{19} \).

Step 1: Identify the Quadrant

Given that \( \cos x = \frac{\sqrt{3}}{5} \) and \( \frac{3\pi}{2} < x < 2\pi \), \( x \) is in the fourth quadrant. In the fourth quadrant, \( \cos x \) is positive and \( \sin x \) is negative.

Step 2: Calculate \( \sin x \) using Pythagorean Identity

Use the identity \( \sin^2 x + \cos^2 x = 1 \). Substitute \( \cos x = \frac{\sqrt{3}}{5} \):\[\left(\frac{\sqrt{3}}{5}\right)^2 + \sin^2 x = 1\]\[\frac{3}{25} + \sin^2 x = 1\]\[\sin^2 x = 1 - \frac{3}{25} = \frac{22}{25}\]\[\sin x = -\sqrt{\frac{22}{25}} = -\frac{\sqrt{22}}{5}\] since the sine is negative in the fourth quadrant.

Step 3: Use Double Angle Formula for \( \cos 2x \)

Apply the double angle formula: \( \cos 2x = 2\cos^2 x - 1 \). Substitute \( \cos x = \frac{\sqrt{3}}{5} \):\[\cos 2x = 2\left(\frac{\sqrt{3}}{5}\right)^2 - 1\]\[\cos 2x = 2\left(\frac{3}{25}\right) - 1 = \frac{6}{25} - 1 = \frac{6}{25} - \frac{25}{25}\]\[\cos 2x = -\frac{19}{25}\]

Step 4: Use Double Angle Formula for \( \sin 2x \)

Apply the double angle formula: \( \sin 2x = 2\sin x \cos x \). Substitute \( \sin x = -\frac{\sqrt{22}}{5} \) and \( \cos x = \frac{\sqrt{3}}{5} \):\[\sin 2x = 2\left(-\frac{\sqrt{22}}{5}\right) \left(\frac{\sqrt{3}}{5}\right)\]\[\sin 2x = 2\left(-\frac{\sqrt{66}}{25}\right) = -\frac{2\sqrt{66}}{25}\]

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

Use the identity \( \tan 2x = \frac{\sin 2x}{\cos 2x} \). Substitute \( \sin 2x = -\frac{2\sqrt{66}}{25} \) and \( \cos 2x = -\frac{19}{25} \):\[\tan 2x = \frac{-\frac{2\sqrt{66}}{25}}{-\frac{19}{25}} = \frac{2\sqrt{66}}{19}\]

Key Concepts

Double Angle Formulas

Double angle formulas are an important part of solving trigonometric problems. These formulas express trigonometric functions of double angles, such as \(\cos 2x\), \(\sin 2x\), and \(\tan 2x\), in terms of single angle functions like \(\cos x\) and \(\sin x\). These are particularly useful for simplifying expressions or solving equations.

For cosine, the double angle formula is \(\cos 2x = 2\cos^2 x - 1\). This formula helps find the cosine of a double angle using the cosine of the single angle. Similarly, for sine, the formula is \(\sin 2x = 2\sin x \cos x\). This allows calculation of the sine of a double angle.

In the case of tangent, the formula is \(\tan 2x = \frac{2\tan x}{1 - \tan^2 x}\). This formula is useful in expressing tangent values for double angles. In our exercise, this simplifies into \(\tan 2x = \frac{\sin 2x}{\cos 2x}\), once the values are substituted. Mastering these formulas can make solving trigonometric questions much simpler.

Pythagorean Identity

The Pythagorean identity is a fundamental principle in trigonometry. It states that for any angle \(x\), the square of the sine plus the square of the cosine equals 1. This is captured in the equation: \(\sin^2 x + \cos^2 x = 1\). This identity links both sine and cosine functions together and is critical in determining unknown values.

In practice, if you know one of these values, say \(\cos x\), you can easily find \(\sin x\). For instance, from the exercise, given \(\cos x = \frac{\sqrt{3}}{5}\), you can find \(\sin x\) through rearrangement:

• \(\sin^2 x = 1 - \cos^2 x\)
• Substitute \(\cos x\) to find \(\sin x = -\frac{\sqrt{22}}{5}\)

Remember, in certain quadrants, sine or cosine may be negative, as seen here where \(\sin x\) is negative because \(x\) falls in the fourth quadrant.

Trigonometric Functions

Trigonometric functions are the core of trigonometry, describing relationships between the angles and sides of triangles. The primary functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function has unique properties that make them useful in geometry and calculus.

Sine is defined as the ratio of the opposite side to the hypotenuse in a right triangle, while cosine is the ratio of the adjacent side to the hypotenuse. Tangent is the ratio of the opposite side to the adjacent side or equivalently \(\tan x = \frac{\sin x}{\cos x}\).

Understanding these functions aids in solving a variety of problems, such as calculating angle sizes or side lengths when certain values are known. In our specific scenario, knowing \(\cos x\) helped deduce \(\sin x\) and subsequently leverage double angle formulas to find \(\sin 2x\) and \(\tan 2x\).
These functions are foundational tools in mathematics and science.

Fourth Quadrant

The fourth quadrant is one of the four sections of the coordinate plane determined by the axes, specifically where \(x\) lies between \(\frac{3\pi}{2}\) and \(2\pi\). In trigonometry, each quadrant has distinct characteristics about the sign of sine, cosine, and tangent functions.

In the fourth quadrant:

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

Knowing these sign rules helps avoid mistakes, especially when solving problems involving multiple angles. For example, if an angle's cosine is known to be positive, but its sine is negative, this instantly indicates the angle lies in the fourth quadrant.
This understanding makes determining the correct value signs for trigonometric functions straightforward and aligns with problem-solving strategies.