Problem 104 - Precalculus

Question

For the following exercises, find the exact values of a \() \sin (2 x),\) b) \(\cos (2 x),\) and \(c ) \tan (2 x)\) without solving for \(x\) If \(\cos x=\frac{2}{3}, \quad\) and \(x\) is in quadrant I.

Step-by-Step Solution

Answer

\( \sin(2x) = \frac{4\sqrt{5}}{9}, \cos(2x) = -\frac{1}{9}, \tan(2x) = -4\sqrt{5} \).

Step 1: Find \( \sin x \) using the Pythagorean identity

Since \( \cos x = \frac{2}{3} \) and \( x \) is in quadrant I, we can use the Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \]Substitute \( \cos x = \frac{2}{3} \) into the equation:\[ \sin^2 x + \left( \frac{2}{3} \right)^2 = 1 \]\[ \sin^2 x + \frac{4}{9} = 1 \]\[ \sin^2 x = 1 - \frac{4}{9} \]\[ \sin^2 x = \frac{5}{9} \]Taking the square root, and considering that \( \sin x \) is positive in the first quadrant:\[ \sin x = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]

Step 2: Calculate \( \sin(2x) \) using the double angle identity

The double angle identity for sine is:\[ \sin(2x) = 2 \sin x \cos x \]Substitute \( \sin x = \frac{\sqrt{5}}{3} \) and \( \cos x = \frac{2}{3} \) into the formula:\[ \sin(2x) = 2 \times \frac{\sqrt{5}}{3} \times \frac{2}{3} \]\[ \sin(2x) = \frac{4\sqrt{5}}{9} \]

Step 3: Calculate \( \cos(2x) \) using the double angle identity

The double angle identity for cosine is:\[ \cos(2x) = 2\cos^2 x - 1 \]Substitute \( \cos x = \frac{2}{3} \) into the formula:\[ \cos(2x) = 2 \left( \frac{2}{3} \right)^2 - 1 \]\[ \cos(2x) = 2 \times \frac{4}{9} - 1 \]\[ \cos(2x) = \frac{8}{9} - 1 \]\[ \cos(2x) = \frac{8}{9} - \frac{9}{9} \]\[ \cos(2x) = -\frac{1}{9} \]

Step 4: Calculate \( \tan(2x) \) using sine and cosine results

The formula for tangent of a double angle is:\[ \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \]Use the results from Step 2 and Step 3:\[ \tan(2x) = \frac{\frac{4\sqrt{5}}{9}}{-\frac{1}{9}} \]When you divide these fractions, the \(9\)s cancel out:\[ \tan(2x) = \frac{4\sqrt{5}}{-1} \]\[ \tan(2x) = -4\sqrt{5} \]

Key Concepts

Pythagorean Identity

Understanding the Pythagorean Identity is a fundamental part of working with trigonometric functions. The identity is expressed as \( \sin^2 x + \cos^2 x = 1 \). This equation reflects the Pythagorean Theorem from geometry applied to the unit circle. In other words, if you imagine a right triangle formed by the x and y coordinates on the unit circle, the hypotenuse is always 1.

To use this identity in practice, if you know either \( \sin x \) or \( \cos x \), you can easily find the other. Here's how it works:

• If \( \cos x = \frac{2}{3} \), then substitute \( \cos x \) into the identity: \( \sin^2 x + \left( \frac{2}{3} \right)^2 = 1 \).
• Calculate \( \left( \frac{2}{3} \right)^2 = \frac{4}{9} \). This gives you \( \sin^2 x + \frac{4}{9} = 1 \).
• Solving for \( \sin^2 x \) gives \( \sin^2 x = 1 - \frac{4}{9} = \frac{5}{9} \).
• To find \( \sin x \), take the square root and remember that in quadrant I, \( \sin x \) is positive: \( \sin x = \frac{\sqrt{5}}{3} \).

Double Angle Formulas

Double Angle Formulas are a set of trigonometric identities that help you find the trigonometric function of twice an angle, given the function of the angle itself. They are especially useful in trigonometry for simplifications and solving equations. There are double angle formulas for sine, cosine, and tangent:

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

To apply these formulas, substitute known values. For instance, with \( \sin x = \frac{\sqrt{5}}{3} \) and \( \cos x = \frac{2}{3} \),

• Find \( \sin(2x) \): Use \( \sin(2x) = 2 \times \frac{\sqrt{5}}{3} \times \frac{2}{3} = \frac{4\sqrt{5}}{9} \).
• Find \( \cos(2x) \): Use \( \cos(2x) = 2 \times \left( \frac{2}{3} \right)^2 - 1 = -\frac{1}{9} \).
• Find \( \tan(2x) \): Use \( \tan(2x) = \frac{\sin (2x)}{\cos (2x)} = -4\sqrt{5} \).

Quadrants

Understanding quadrants is critical when working with trigonometric functions, as the sign of the function values depends on the quadrant in which the angle resides. The four quadrants in a circle are divided by the x-axis and y-axis:

• Quadrant I: Both \( \sin x \) and \( \cos x \) are positive.
• Quadrant II: \( \sin x \) is positive, \( \cos x \) is negative.
• Quadrant III: Both \( \sin x \) and \( \cos x \) are negative.
• Quadrant IV: \( \sin x \) is negative, \( \cos x \) is positive.

In the problem, \( x \) is given to be in Quadrant I. This information indicates that both \( \sin x \) and \( \cos x \) must be positive, confirming our solution is correctly using the positive square root of \( \sin^2 x \). Understanding quadrants helps ensure the accuracy of trigonometric calculations and interpretations of the function values.