Problem 9 - Precalculus Mathematics for Calculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. $$\tan x=-\frac{1}{3}, \quad \cos x>0$$

Step-by-Step Solution

Answer

\(\sin 2x = -\frac{3}{5}\), \(\cos 2x = \frac{4}{5}\), \(\tan 2x = -\frac{3}{4}\).

Step 1: Identify Quad I or IV

Since \(\tan x = -\frac{1}{3}\) and \(\cos x > 0\), \(x\) must be in the fourth quadrant where tangent is negative and cosine is positive.

Step 2: Use Pythagorean Identity

For tangent, \(\tan x = \frac{\sin x}{\cos x}\). Given \(\tan x = -\frac{1}{3}\), this means \(\sin x = -\frac{1}{3} \cos x\).Using \(\sin^2 x + \cos^2 x = 1\), substitute \(\sin x\) and solve for \(\cos x\):\(\left(-\frac{1}{3}\cos x\right)^2 + \cos^2 x = 1\)\(\frac{1}{9}\cos^2 x + \cos^2 x = 1\)\(\frac{10}{9}\cos^2 x = 1\)\(\cos^2 x = \frac{9}{10}\)\(\cos x = \frac{3\sqrt{10}}{10}\).

Step 3: Find \(\sin x\)

Using \(\sin x = -\frac{1}{3} \cos x\), we find \(\sin x = -\frac{1}{3} \times \frac{3\sqrt{10}}{10} = -\frac{\sqrt{10}}{10}\).

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

Use the double angle formula for sine: \(\sin 2x = 2 \sin x \cos x\):\(\sin 2x = 2\left(-\frac{\sqrt{10}}{10}\right) \left(\frac{3\sqrt{10}}{10}\right) \)\(= 2 \times \frac{-3 \times 10}{100} \)\(= -\frac{6}{10} = -\frac{3}{5}\).

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

Use the double angle formula for cosine: \(\cos 2x = \cos^2 x - \sin^2 x\).\(\cos 2x = \frac{9}{10} - \left(-\frac{\sqrt{10}}{10}\right)^2\)\(= \frac{9}{10} - \frac{10}{100}\)\(= \frac{9}{10} - \frac{1}{10} = \frac{8}{10} = \frac{4}{5}\).

Step 6: Calculate \(\tan 2x\)

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

Key Concepts

Tangent of an Angle

The tangent of an angle, represented as \( \tan x \), describes the ratio of the sine and cosine of that angle. It can often help us determine important properties of the angle when combined with the knowledge of which quadrant it resides. In particular, knowing the sign of \( \tan x \) aids in finding the correct quadrant, as tangent is positive in the first and third quadrants and negative in the second and fourth.

• Formula: \( \tan x = \frac{\sin x}{\cos x} \)
• In the given example, \( \tan x = -\frac{1}{3} \), and since \( \cos x > 0 \), x is located in the fourth quadrant.

This helps us use the tangent to extract information about sine and cosine even when we do not have their explicit values yet.

Cosine of an Angle

The cosine of an angle, denoted as \( \cos x \), gives you the horizontal coordinate of a point on the unit circle corresponding to angle \( x \). In trigonometry, cosine is fundamental in forming identities used to compute values like \( \cos 2x \) through double angle formulas.

• Key identity: \( \cos^2 x + \sin^2 x = 1 \)
• In our scenario, knowing \( \tan x \) allows us to determine \( \cos x = \frac{3\sqrt{10}}{10} \).

This value of \( \cos x \) is crucial in computing the double angle, \( \cos 2x \), using the formula \( \cos 2x = \cos^2 x - \sin^2 x \). An important takeaway here is how interconnected the trigonometric functions are, with each one helping to find the others.

Sine of an Angle

Sine, represented as \( \sin x \), can be understood as giving the vertical coordinate on the unit circle for a given angle \( x \). Like cosine, it is key to solving trigonometric problems and is involved in many identities and formulas. Using the tangent and Pythagorean identity, you can solve for \( \sin x \) when \( \tauan x \) and \( \cos x \) are known.

• Find \( \sin x \) with: \( \sin x = -\frac{1}{3} \cos x \)
• Calculated in our problem as \( \sin x = -\frac{\sqrt{10}}{10} \)

Once we have \( \sin x \), the double angle formula \( \sin 2x = 2 \sin x \cos x \) can be applied to find the sine of the doubled angle. Double angle formulas are powerful tools that allow us to derive new relationships and calculations essential in various applications of trigonometry in mathematics and physics.