What Is The Exact Value Of $\sec \frac{3 \pi}{4}$?A. $-\frac{2 \sqrt{3}}{3}$ B. $-\frac{\sqrt{2}}{2}$ C. $-2$ D. $-\sqrt{2}$

What is the Exact Value of $\sec \frac{3 \pi}{4}$?

Understanding the Problem

The problem requires finding the exact value of the secant of $\frac{3 \pi}{4}$, which is a trigonometric function. To solve this problem, we need to recall the definition of the secant function and its relationship with the cosine function.

Recalling the Definition of Secant

The secant function is defined as the reciprocal of the cosine function. In other words, $\sec \theta = \frac{1}{\cos \theta}$. This means that to find the value of $\sec \theta$, we need to find the value of $\cos \theta$ and then take its reciprocal.

Finding the Value of $\cos \frac{3 \pi}{4}$

To find the value of $\cos \frac{3 \pi}{4}$, we need to recall the unit circle and the values of cosine for common angles. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The cosine of an angle is the x-coordinate of the point on the unit circle corresponding to that angle.

Using the Unit Circle

The angle $\frac{3 \pi}{4}$ is in the second quadrant of the unit circle. In the second quadrant, the x-coordinate is negative. The reference angle for $\frac{3 \pi}{4}$ is $\frac{\pi}{4}$, which is a common angle with known cosine and sine values.

Recalling the Values of Cosine and Sine for $\frac{\pi}{4}$

The values of cosine and sine for $\frac{\pi}{4}$ are:

• $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$
• $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

Finding the Value of $\cos \frac{3 \pi}{4}$

Since the angle $\frac{3 \pi}{4}$ is in the second quadrant, the x-coordinate is negative. Therefore, the value of $\cos \frac{3 \pi}{4}$ is:

• $\cos \frac{3 \pi}{4} = -\frac{\sqrt{2}}{2}$

Finding the Value of $\sec \frac{3 \pi}{4}$

Now that we have found the value of $\cos \frac{3 \pi}{4}$, we can find the value of $\sec \frac{3 \pi}{4}$ by taking the reciprocal of $\cos \frac{3 \pi}{4}$:

• $\sec \frac{3 \pi}{4} = \frac{1}{\cos \frac{3 \pi}{4}} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\frac{2\sqrt{2}}{2} = -\sqrt{2}$

Conclusion

The exact value of $\sec \frac{3 \pi}{4}$ is $-\sqrt{2}$. This value can be verified by using a calculator or by graphing the secant function.

Answer

The correct answer is D. $-\sqrt{2}$.

Common Mistakes

• Some students may confuse the value of $\cos \frac{3 \pi}{4}$ with the value of $\sin \frac{3 \pi}{4}$.
• Some students may forget to take the reciprocal of $\cos \frac{3 \pi}{4}$ to find the value of $\sec \frac{3 \pi}{4}$.

Tips and Tricks

• To find the value of $\sec \theta$, recall that it is the reciprocal of $\cos \theta$.
• Use the unit circle to find the value of $\cos \theta$.
• Be careful when taking the reciprocal of a negative value.

Real-World Applications

• The secant function is used in many real-world applications, such as physics, engineering, and computer science.
• The secant function is used to model the behavior of objects in motion, such as the trajectory of a projectile.

Practice Problems

• Find the value of $\sec \frac{5 \pi}{4}$.
• Find the value of $\sec \frac{7 \pi}{4}$.

Solutions

• $\sec \frac{5 \pi}{4} = -\frac{2\sqrt{2}}{2} = -\sqrt{2}$
• $\sec \frac{7 \pi}{4} = -\frac{2\sqrt{2}}{2} = -\sqrt{2}$
  Q&A: Secant Function

Frequently Asked Questions

Q: What is the secant function?

A: The secant function is a trigonometric function that is defined as the reciprocal of the cosine function. In other words, $\sec \theta = \frac{1}{\cos \theta}$.

Q: How do I find the value of $\sec \theta$?

A: To find the value of $\sec \theta$, you need to find the value of $\cos \theta$ and then take its reciprocal. You can use the unit circle to find the value of $\cos \theta$.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The cosine of an angle is the x-coordinate of the point on the unit circle corresponding to that angle.

Q: How do I use the unit circle to find the value of $\cos \theta$?

A: To use the unit circle to find the value of $\cos \theta$, you need to identify the quadrant in which the angle $\theta$ lies. Then, you can use the reference angle to find the value of $\cos \theta$.

Q: What is a reference angle?

A: A reference angle is an angle that has the same cosine value as the original angle. For example, the reference angle for $\frac{3 \pi}{4}$ is $\frac{\pi}{4}$.

Q: How do I find the value of $\sec \frac{3 \pi}{4}$?

A: To find the value of $\sec \frac{3 \pi}{4}$, you need to find the value of $\cos \frac{3 \pi}{4}$ and then take its reciprocal. The value of $\cos \frac{3 \pi}{4}$ is $-\frac{\sqrt{2}}{2}$. Therefore, the value of $\sec \frac{3 \pi}{4}$ is $-\frac{2}{\sqrt{2}} = -\sqrt{2}$.

Q: What are some common mistakes to avoid when working with the secant function?

A: Some common mistakes to avoid when working with the secant function include confusing the value of $\cos \theta$ with the value of $\sin \theta$, and forgetting to take the reciprocal of $\cos \theta$ to find the value of $\sec \theta$.

Q: What are some real-world applications of the secant function?

A: The secant function is used in many real-world applications, such as physics, engineering, and computer science. It is used to model the behavior of objects in motion, such as the trajectory of a projectile.

Q: How do I practice working with the secant function?

A: You can practice working with the secant function by finding the values of $\sec \theta$ for different angles, and by using the unit circle to find the values of $\cos \theta$.

Q: What are some tips and tricks for working with the secant function?

A: Some tips and tricks for working with the secant function include using the unit circle to find the values of $\cos \theta$, and being careful when taking the reciprocal of $\cos \theta$ to find the value of $\sec \theta$.

Q: What are some common formulas and identities involving the secant function?

A: Some common formulas and identities involving the secant function include:

• $\sec \theta = \frac{1}{\cos \theta}$
• $\csc \theta = \frac{1}{\sin \theta}$
• $\sec^2 \theta - \tan^2 \theta = 1$

Q: How do I use the secant function to solve problems in physics and engineering?

A: The secant function is used to model the behavior of objects in motion, such as the trajectory of a projectile. To use the secant function to solve problems in physics and engineering, you need to identify the angle and the velocity of the object, and then use the secant function to find the position and velocity of the object at different times.

Q: What are some common mistakes to avoid when using the secant function to solve problems in physics and engineering?

A: Some common mistakes to avoid when using the secant function to solve problems in physics and engineering include forgetting to take the reciprocal of $\cos \theta$ to find the value of $\sec \theta$, and not using the correct units when working with the secant function.

Q: How do I use the secant function to solve problems in computer science?

A: The secant function is used in computer science to model the behavior of algorithms and data structures. To use the secant function to solve problems in computer science, you need to identify the angle and the velocity of the algorithm or data structure, and then use the secant function to find the position and velocity of the algorithm or data structure at different times.

Q: What are some common mistakes to avoid when using the secant function to solve problems in computer science?

A: Some common mistakes to avoid when using the secant function to solve problems in computer science include forgetting to take the reciprocal of $\cos \theta$ to find the value of $\sec \theta$, and not using the correct units when working with the secant function.