Solve For Θ \theta Θ .1. Sin ⁡ 2 ( Θ ) − Cos ⁡ 2 ( Θ ) = − 1 \sin^2(\theta) - \cos^2(\theta) = -1 Sin 2 ( Θ ) − Cos 2 ( Θ ) = − 1 Which Of The Following Is True For Θ \theta Θ ?A. Θ = 2 K Π \theta = 2k\pi Θ = 2 Kπ , Where K K K Is An IntegerB. Θ = Π 2 + 2 K Π \theta = \frac{\pi}{2} + 2k\pi Θ = 2 Π ​ + 2 Kπ , Where

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Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving a specific trigonometric equation involving sine and cosine functions. We will use various trigonometric identities and properties to simplify the equation and find the solution.

The Given Equation

The given equation is sin⁡2(θ)−cos⁡2(θ)=−1\sin^2(\theta) - \cos^2(\theta) = -1. This equation involves the square of sine and cosine functions, and we need to find the value of θ\theta that satisfies this equation.

Using Trigonometric Identities

To solve this equation, we can use the Pythagorean identity, which states that sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1. We can rewrite the given equation as:

sin⁡2(θ)−cos⁡2(θ)=−1\sin^2(\theta) - \cos^2(\theta) = -1

sin⁡2(θ)+cos⁡2(θ)−2cos⁡2(θ)=−1\sin^2(\theta) + \cos^2(\theta) - 2\cos^2(\theta) = -1

1−2cos⁡2(θ)=−11 - 2\cos^2(\theta) = -1

−2cos⁡2(θ)=−2-2\cos^2(\theta) = -2

cos⁡2(θ)=1\cos^2(\theta) = 1

cos⁡(θ)=±1\cos(\theta) = \pm 1

Solving for θ\theta

Now that we have simplified the equation, we can solve for θ\theta. We know that cos⁡(θ)=±1\cos(\theta) = \pm 1, which means that θ\theta can take on values that make cos⁡(θ)\cos(\theta) equal to 11 or −1-1.

Case 1: cos⁡(θ)=1\cos(\theta) = 1

When cos⁡(θ)=1\cos(\theta) = 1, we know that θ\theta is an integer multiple of 2π2\pi, i.e., θ=2kπ\theta = 2k\pi, where kk is an integer.

Case 2: cos⁡(θ)=−1\cos(\theta) = -1

When cos⁡(θ)=−1\cos(\theta) = -1, we know that θ\theta is an odd integer multiple of π\pi, i.e., θ=π2+2kπ\theta = \frac{\pi}{2} + 2k\pi, where kk is an integer.

Conclusion

In conclusion, we have solved the given trigonometric equation and found the values of θ\theta that satisfy the equation. We used the Pythagorean identity to simplify the equation and then solved for θ\theta using the properties of cosine function.

Answer

The correct answer is:

B. θ=π2+2kπ\theta = \frac{\pi}{2} + 2k\pi, where kk is an integer

Discussion

This problem requires a deep understanding of trigonometric functions and their properties. The Pythagorean identity is a fundamental concept in trigonometry, and using it to simplify the equation is a crucial step in solving the problem. The solution involves identifying the values of θ\theta that make cos⁡(θ)\cos(\theta) equal to 11 or −1-1, and then expressing these values in terms of integer multiples of 2π2\pi and π\pi.

Additional Resources

For more information on trigonometric equations and identities, please refer to the following resources:

Related Problems

References

Introduction

In our previous article, we solved a trigonometric equation involving sine and cosine functions. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving trigonometric equations.

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions such as sine, cosine, and tangent. These equations can be used to model real-world problems and can be solved using various techniques and identities.

Q: What are the common trigonometric identities?

A: The common trigonometric identities include:

  • Pythagorean identity: sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1
  • Sum and difference identities: sin⁡(A+B)=sin⁡(A)cos⁡(B)+cos⁡(A)sin⁡(B)\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B) and sin⁡(A−B)=sin⁡(A)cos⁡(B)−cos⁡(A)sin⁡(B)\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B)
  • Double angle identities: sin⁡(2θ)=2sin⁡(θ)cos⁡(θ)\sin(2\theta) = 2\sin(\theta)\cos(\theta) and cos⁡(2θ)=2cos⁡2(θ)−1\cos(2\theta) = 2\cos^2(\theta) - 1
  • Half angle identities: sin⁡(θ2)=±1−cos⁡(θ)2\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1-\cos(\theta)}{2}} and cos⁡(θ2)=±1+cos⁡(θ)2\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1+\cos(\theta)}{2}}

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can use the following steps:

  1. Simplify the equation using trigonometric identities.
  2. Isolate the trigonometric function.
  3. Use the properties of the trigonometric function to find the solution.

Q: What are the common trigonometric functions?

A: The common trigonometric functions include:

  • Sine: sin⁡(θ)\sin(\theta)
  • Cosine: cos⁡(θ)\cos(\theta)
  • Tangent: tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
  • Cotangent: cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
  • Secant: sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}
  • Cosecant: csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}

Q: How do I use the Pythagorean identity?

A: The Pythagorean identity is used to simplify trigonometric equations. It states that sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1. You can use this identity to simplify equations involving sine and cosine functions.

Q: How do I use the sum and difference identities?

A: The sum and difference identities are used to simplify trigonometric equations involving the sum and difference of two angles. They state that sin⁡(A+B)=sin⁡(A)cos⁡(B)+cos⁡(A)sin⁡(B)\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B) and sin⁡(A−B)=sin⁡(A)cos⁡(B)−cos⁡(A)sin⁡(B)\sin(A-B) = \sin(A)\cos(B) - \cos(A)\sin(B).

Q: How do I use the double angle identities?

A: The double angle identities are used to simplify trigonometric equations involving the double angle. They state that sin⁡(2θ)=2sin⁡(θ)cos⁡(θ)\sin(2\theta) = 2\sin(\theta)\cos(\theta) and cos⁡(2θ)=2cos⁡2(θ)−1\cos(2\theta) = 2\cos^2(\theta) - 1.

Q: How do I use the half angle identities?

A: The half angle identities are used to simplify trigonometric equations involving the half angle. They state that sin⁡(θ2)=±1−cos⁡(θ)2\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1-\cos(\theta)}{2}} and cos⁡(θ2)=±1+cos⁡(θ)2\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1+\cos(\theta)}{2}}.

Conclusion

In conclusion, solving trigonometric equations requires a deep understanding of trigonometric functions and their properties. By using the common trigonometric identities and techniques, you can simplify and solve trigonometric equations. We hope this Q&A guide has helped you understand the concepts and techniques involved in solving trigonometric equations.

Additional Resources

For more information on trigonometric equations and identities, please refer to the following resources:

Related Problems

References