Given That Tan ⁡ Θ = − 1 \tan \theta = -1 Tan Θ = − 1 , What Is The Value Of Sec ⁡ Θ \sec \theta Sec Θ For 3 Π 2 \textless Θ \textless 2 Π \frac{3\pi}{2} \ \textless \ \theta \ \textless \ 2\pi 2 3 Π ​ \textless Θ \textless 2 Π ?A. − 2 -\sqrt{2} − 2 ​ B. 2 \sqrt{2} 2 ​ C. 0 D. 1

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Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the relationships between different trigonometric functions. In this article, we will focus on solving a specific trigonometric equation involving the secant function, given that tan⁡θ=−1\tan \theta = -1 and 3π2 \textless θ \textless 2π\frac{3\pi}{2} \ \textless \ \theta \ \textless \ 2\pi. We will explore the properties of the secant function, use trigonometric identities to simplify the equation, and finally, determine the value of sec⁡θ\sec \theta.

Understanding the Secant Function

The secant function is the reciprocal of the cosine function, denoted as sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}. It is an essential function in trigonometry, and its value can be determined using the cosine function. In this case, we are given that tan⁡θ=−1\tan \theta = -1, and we need to find the value of sec⁡θ\sec \theta.

Using Trigonometric Identities

To solve this equation, we can use the trigonometric identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}. Since we are given that tan⁡θ=−1\tan \theta = -1, we can write sin⁡θcos⁡θ=−1\frac{\sin \theta}{\cos \theta} = -1. This implies that sin⁡θ=−cos⁡θ\sin \theta = -\cos \theta.

Simplifying the Equation

Using the identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, we can substitute sin⁡θ=−cos⁡θ\sin \theta = -\cos \theta into the equation. This gives us (−cos⁡θ)2+cos⁡2θ=1(-\cos \theta)^2 + \cos^2 \theta = 1. Simplifying this equation, we get 2cos⁡2θ=12\cos^2 \theta = 1.

Solving for cos⁡θ\cos \theta

Dividing both sides of the equation by 2, we get cos⁡2θ=12\cos^2 \theta = \frac{1}{2}. Taking the square root of both sides, we get cos⁡θ=±12=±12\cos \theta = \pm \sqrt{\frac{1}{2}} = \pm \frac{1}{\sqrt{2}}. Since θ\theta lies in the fourth quadrant, where cosine is negative, we take the negative value, cos⁡θ=−12\cos \theta = -\frac{1}{\sqrt{2}}.

Finding the Value of sec⁡θ\sec \theta

Now that we have found the value of cos⁡θ\cos \theta, we can determine the value of sec⁡θ\sec \theta. Using the definition of the secant function, sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, we can substitute the value of cos⁡θ\cos \theta into the equation. This gives us sec⁡θ=1−12=−2\sec \theta = \frac{1}{-\frac{1}{\sqrt{2}}} = -\sqrt{2}.

Conclusion

In this article, we have solved a trigonometric equation involving the secant function, given that tan⁡θ=−1\tan \theta = -1 and 3π2 \textless θ \textless 2π\frac{3\pi}{2} \ \textless \ \theta \ \textless \ 2\pi. We have used trigonometric identities to simplify the equation, and finally, determined the value of sec⁡θ\sec \theta. The value of sec⁡θ\sec \theta is −2-\sqrt{2}.

Final Answer

The final answer is −2\boxed{-\sqrt{2}}.

Discussion

This problem requires a deep understanding of trigonometric functions and their relationships. The use of trigonometric identities is essential in solving this equation. The value of sec⁡θ\sec \theta can be determined using the definition of the secant function and the value of cos⁡θ\cos \theta. This problem is a great example of how trigonometric functions can be used to solve real-world problems.

Additional Resources

For more information on trigonometric functions and their relationships, please refer to the following resources:

Related Problems

Conclusion

In conclusion, solving trigonometric equations requires a deep understanding of trigonometric functions and their relationships. The use of trigonometric identities is essential in solving these equations. The value of sec⁡θ\sec \theta can be determined using the definition of the secant function and the value of cos⁡θ\cos \theta. This problem is a great example of how trigonometric functions can be used to solve real-world problems.

Introduction

In our previous article, we solved a trigonometric equation involving the secant function, given that tan⁡θ=−1\tan \theta = -1 and 3π2 \textless θ \textless 2π\frac{3\pi}{2} \ \textless \ \theta \ \textless \ 2\pi. We used trigonometric identities to simplify the equation and determined the value of sec⁡θ\sec \theta. In this article, we will answer some frequently asked questions related to solving trigonometric equations.

Q: What is the difference between the secant and cosine functions?

A: The secant function is the reciprocal of the cosine function, denoted as sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}. This means that the secant function is equal to 1 divided by the cosine function.

Q: How do I use trigonometric identities to simplify trigonometric equations?

A: Trigonometric identities are essential in simplifying trigonometric equations. You can use identities such as sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 and tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} to simplify the equation.

Q: What is the value of sec⁡θ\sec \theta when tan⁡θ=−1\tan \theta = -1 and 3π2 \textless θ \textless 2π\frac{3\pi}{2} \ \textless \ \theta \ \textless \ 2\pi?

A: The value of sec⁡θ\sec \theta is −2-\sqrt{2} when tan⁡θ=−1\tan \theta = -1 and 3π2 \textless θ \textless 2π\frac{3\pi}{2} \ \textless \ \theta \ \textless \ 2\pi.

Q: How do I determine the value of sec⁡θ\sec \theta when cos⁡θ\cos \theta is negative?

A: When cos⁡θ\cos \theta is negative, you can use the definition of the secant function, sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, to determine the value of sec⁡θ\sec \theta. Since the secant function is the reciprocal of the cosine function, the value of sec⁡θ\sec \theta will be negative when cos⁡θ\cos \theta is negative.

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

Q: How do I use trigonometric identities to solve trigonometric equations?

A: To use trigonometric identities to solve trigonometric equations, you can start by simplifying the equation using the identities. Then, you can use the simplified equation to determine the value of the trigonometric function.

Q: What are some tips for solving trigonometric equations?

A: Some tips for solving trigonometric equations include:

  • Use trigonometric identities to simplify the equation
  • Determine the value of the trigonometric function using the simplified equation
  • Check your answer to make sure it is correct

Conclusion

In conclusion, solving trigonometric equations requires a deep understanding of trigonometric functions and their relationships. The use of trigonometric identities is essential in simplifying the equation and determining the value of the trigonometric function. By following the tips and using the common trigonometric identities, you can solve trigonometric equations with ease.

Final Answer

The final answer is −2\boxed{-\sqrt{2}}.

Discussion

This problem requires a deep understanding of trigonometric functions and their relationships. The use of trigonometric identities is essential in simplifying the equation and determining the value of the trigonometric function. By following the tips and using the common trigonometric identities, you can solve trigonometric equations with ease.

Additional Resources

For more information on trigonometric functions and their relationships, please refer to the following resources:

Related Problems

Conclusion

In conclusion, solving trigonometric equations requires a deep understanding of trigonometric functions and their relationships. The use of trigonometric identities is essential in simplifying the equation and determining the value of the trigonometric function. By following the tips and using the common trigonometric identities, you can solve trigonometric equations with ease.