Rewrite The Expression In Terms Of The Given Angle's Reference Angle; Then Evaluate The Result. Write The Exact Answer. Do Not Round.$\tan \left(\frac{23 \pi}{3}\right$\]

Introduction

In trigonometry, the tangent function is a fundamental concept that deals with the ratio of the opposite side to the adjacent side in a right-angled triangle. When dealing with angles in standard position, it's essential to understand how to rewrite expressions in terms of the given angle's reference angle. This article will guide you through the process of rewriting the expression $\tan \left(\frac{23 \pi}{3}\right)$ in terms of the given angle's reference angle and then evaluate the result.

Understanding Reference Angles

A reference angle is the acute angle between the terminal side of an angle and the x-axis. It's a crucial concept in trigonometry as it helps us find the values of trigonometric functions for any angle. To find the reference angle, we need to determine the quadrant in which the angle lies and then find the acute angle between the terminal side and the x-axis.

Rewriting the Expression

To rewrite the expression $\tan \left(\frac{23 \pi}{3}\right)$ in terms of the given angle's reference angle, we need to first find the reference angle. The given angle is $\frac{23 \pi}{3}$, which is in the third quadrant. To find the reference angle, we can subtract the nearest multiple of $2\pi$ from the given angle.

$\frac{23 \pi}{3} - 2\pi = \frac{23 \pi}{3} - \frac{6\pi}{3} = \frac{17 \pi}{3}$

The reference angle is $\frac{17 \pi}{3}$.

Rewriting the Expression in Terms of the Reference Angle

Now that we have the reference angle, we can rewrite the expression $\tan \left(\frac{23 \pi}{3}\right)$ in terms of the reference angle. Since the given angle is in the third quadrant, the tangent function will be negative.

$\tan \left(\frac{23 \pi}{3}\right) = -\tan \left(\frac{17 \pi}{3}\right)$

Evaluating the Result

To evaluate the result, we need to find the value of $\tan \left(\frac{17 \pi}{3}\right)$. Since the reference angle is $\frac{17 \pi}{3}$, we can use the unit circle or trigonometric identities to find the value of the tangent function.

$\tan \left(\frac{17 \pi}{3}\right) = \tan \left(2\pi + \frac{5\pi}{3}\right)$

Using the identity $\tan (2\pi + \theta) = \tan \theta$, we get:

$\tan \left(\frac{17 \pi}{3}\right) = \tan \left(\frac{5\pi}{3}\right)$

The value of $\tan \left(\frac{5\pi}{3}\right)$ is:

$\tan \left(\frac{5\pi}{3}\right) = -\frac{1}{\sqrt{3}}$

Now that we have the value of $\tan \left(\frac{17 \pi}{3}\right)$, we can substitute it back into the expression:

$\tan \left(\frac{23 \pi}{3}\right) = -\tan \left(\frac{17 \pi}{3}\right) = -\left(-\frac{1}{\sqrt{3}}\right)$

Simplifying the expression, we get:

$\tan \left(\frac{23 \pi}{3}\right) = \frac{1}{\sqrt{3}}$

Conclusion

Q: What is the reference angle in trigonometry?

A: The reference angle is the acute angle between the terminal side of an angle and the x-axis. It's a crucial concept in trigonometry as it helps us find the values of trigonometric functions for any angle.

Q: How do I find the reference angle?

A: To find the reference angle, you need to determine the quadrant in which the angle lies and then find the acute angle between the terminal side and the x-axis. You can use the unit circle or trigonometric identities to find the reference angle.

Q: What is the difference between the given angle and the reference angle?

A: The given angle is the angle in standard position, while the reference angle is the acute angle between the terminal side of the given angle and the x-axis. The reference angle is always less than or equal to the given angle.

Q: How do I rewrite an expression in terms of the reference angle?

A: To rewrite an expression in terms of the reference angle, you need to first find the reference angle. Then, you can use the reference angle to rewrite the expression. For example, if the given angle is $\frac{23 \pi}{3}$, you can rewrite the expression as $-\tan \left(\frac{17 \pi}{3}\right)$.

Q: What is the significance of the reference angle in trigonometry?

A: The reference angle is a crucial concept in trigonometry as it helps us find the values of trigonometric functions for any angle. It's used to rewrite expressions in terms of the given angle's reference angle, which makes it easier to evaluate the result.

Q: Can I use the reference angle to find the values of trigonometric functions?

A: Yes, you can use the reference angle to find the values of trigonometric functions. By using the reference angle, you can rewrite expressions in terms of the given angle's reference angle, which makes it easier to evaluate the result.

Q: What are some common mistakes to avoid when working with reference angles?

A: Some common mistakes to avoid when working with reference angles include:

• Not determining the quadrant in which the angle lies
• Not finding the acute angle between the terminal side and the x-axis
• Not using the correct trigonometric identities to find the reference angle
• Not rewriting the expression in terms of the reference angle correctly

Q: How do I apply the concept of reference angles in real-world problems?

A: The concept of reference angles is used in various real-world problems, such as:

• Calculating the height of a building or a mountain
• Determining the distance between two points on a map
• Finding the angle of elevation or depression in a surveying problem
• Calculating the force or torque in a mechanical system

Q: What are some advanced topics in trigonometry that involve reference angles?

A: Some advanced topics in trigonometry that involve reference angles include:

• Double-angle and half-angle formulas
• Sum and difference formulas
• Trigonometric identities and equations
• Polar coordinates and complex numbers

Q: How do I practice and improve my skills in working with reference angles?

A: To practice and improve your skills in working with reference angles, you can:

• Practice finding reference angles for different angles
• Rewrite expressions in terms of the reference angle
• Use trigonometric identities to find the values of trigonometric functions
• Apply the concept of reference angles to real-world problems
• Review and practice regularly to build your skills and confidence.