Angle \[$ X \$\] Is A Third Quadrant Angle Such That \[$ \cos X = -\frac{2}{3} \$\].What Is The Exact Value Of \[$ \cos \left(\frac{x}{2}\right) \$\]?Enter Your Answer, In Simplest Radical Form, In The Box.$\[ \cos

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Introduction

In trigonometry, angles in the third quadrant have a cosine value that is negative. Given an angle { x $}$ in the third quadrant, where { \cos x = -\frac{2}{3} $}$, we are tasked with finding the exact value of { \cos \left(\frac{x}{2}\right) $}$. This problem requires us to apply trigonometric identities and half-angle formulas to solve for the desired value.

Recall of Trigonometric Identities

Before we proceed, let's recall some essential trigonometric identities that will be useful in solving this problem.

  • Pythagorean Identity: { \sin^2 x + \cos^2 x = 1 $}$
  • Half-Angle Formula for Cosine: { \cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} $}$

Step 1: Find the Value of { \sin x $}$

To find the value of { \sin x $}$, we can use the Pythagorean identity. Given that { \cos x = -\frac{2}{3} $}$, we can substitute this value into the Pythagorean identity to solve for { \sin x $}$.

{ \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 of both sides, we get:

{ \sin x = \pm \sqrt{\frac{5}{9}} $}$

Since { x $}$ is in the third quadrant, where sine values are negative, we take the negative square root:

{ \sin x = -\sqrt{\frac{5}{9}} $}$

Step 2: Apply the Half-Angle Formula for Cosine

Now that we have the values of { \cos x $}$ and { \sin x $}$, we can apply the half-angle formula for cosine to find the value of { \cos \left(\frac{x}{2}\right) $}$.

{ \cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} $}$

Substituting the value of { \cos x = -\frac{2}{3} $}$ into the formula, we get:

{ \cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \frac{2}{3}}{2}} $}$

{ \cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{\frac{1}{3}}{2}} $}$

{ \cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1}{6}} $}$

Simplifying the expression, we get:

{ \cos \left(\frac{x}{2}\right) = \pm \frac{1}{\sqrt{6}} $}$

Rationalizing the denominator, we get:

{ \cos \left(\frac{x}{2}\right) = \pm \frac{\sqrt{6}}{6} $}$

Since { x $}$ is in the third quadrant, where cosine values are negative, we take the negative value:

{ \cos \left(\frac{x}{2}\right) = -\frac{\sqrt{6}}{6} $}$

Conclusion

Q: What is the Pythagorean identity in trigonometry?

A: The Pythagorean identity is a fundamental concept in trigonometry that states: { \sin^2 x + \cos^2 x = 1 $}$. This identity is used to relate the sine and cosine values of an angle.

Q: How do I use the Pythagorean identity to find the value of { \sin x $}$ given the value of { \cos x $}$?

A: To find the value of { \sin x $}$ given the value of { \cos x $}$, you can use the Pythagorean identity: { \sin^2 x + \cos^2 x = 1 $}$. Rearrange the equation to isolate { \sin^2 x $}$: { \sin^2 x = 1 - \cos^2 x $}$. Then, take the square root of both sides to find the value of { \sin x $}$.

Q: What is the half-angle formula for cosine?

A: The half-angle formula for cosine is: { \cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} $}$. This formula is used to find the value of the cosine of a half-angle given the value of the cosine of the original angle.

Q: How do I apply the half-angle formula for cosine to find the value of { \cos \left(\frac{x}{2}\right) $}$ given the value of { \cos x $}$?

A: To apply the half-angle formula for cosine, substitute the value of { \cos x $}$ into the formula: { \cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} $}$. Simplify the expression to find the value of { \cos \left(\frac{x}{2}\right) $}$.

Q: What is the significance of the { \pm $}$ sign in the half-angle formula for cosine?

A: The { \pm $}$ sign in the half-angle formula for cosine indicates that the value of { \cos \left(\frac{x}{2}\right) $}$ can be either positive or negative, depending on the value of { \cos x $}$.

Q: How do I determine whether the value of { \cos \left(\frac{x}{2}\right) $}$ is positive or negative?

A: To determine whether the value of { \cos \left(\frac{x}{2}\right) $}$ is positive or negative, you need to consider the quadrant in which the angle { x $}$ lies. If the angle { x $}$ lies in the third quadrant, where cosine values are negative, then the value of { \cos \left(\frac{x}{2}\right) $}$ will be negative.

Q: What are some common applications of trigonometric identities and half-angle formulas?

A: Trigonometric identities and half-angle formulas have numerous applications in various fields, including physics, engineering, and mathematics. Some common applications include:

  • Calculating the values of trigonometric functions for specific angles
  • Solving trigonometric equations and inequalities
  • Finding the values of trigonometric functions for half-angles and quarter-angles
  • Applying trigonometric identities to simplify complex expressions

Conclusion

In this article, we have discussed some frequently asked questions about trigonometric identities and half-angle formulas. We have covered topics such as the Pythagorean identity, the half-angle formula for cosine, and the significance of the { \pm $}$ sign in the half-angle formula for cosine. We have also provided examples and explanations to help illustrate the concepts and formulas.