Select All Angle Measures For Which Sin ⁡ Θ = − 2 2 \sin \theta = -\frac{\sqrt{2}}{2} Sin Θ = − 2 2 ​ ​ .A. 3 Π 4 \frac{3 \pi}{4} 4 3 Π ​ B. 5 Π 4 \frac{5 \pi}{4} 4 5 Π ​ C. 7 Π 4 \frac{7 \pi}{4} 4 7 Π ​ D. 9 Π 4 \frac{9 \pi}{4} 4 9 Π ​ E. 13 Π 4 \frac{13 \pi}{4} 4 13 Π ​

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Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the relationships between the trigonometric functions. In this article, we will focus on solving the equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2}, which is a classic example of a trigonometric equation. We will explore the different methods of solving this equation and provide a step-by-step guide to finding the angle measures.

Understanding the Sine Function

Before we dive into solving the equation, it's essential to understand the sine function. The sine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function has a period of 2π2\pi, which means that it repeats every 2π2\pi radians.

The Equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2}

The equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2} is a simple trigonometric equation that can be solved using various methods. We will use the unit circle to visualize the solution.

Using the Unit Circle

The unit circle is a fundamental concept in trigonometry, and it is used to visualize the sine and cosine functions. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The sine and cosine functions are defined as the ratios of the coordinates of a point on the unit circle to the radius of the circle.

To solve the equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2}, we need to find the angle measures that correspond to the sine value of −22-\frac{\sqrt{2}}{2}. We can use the unit circle to visualize the solution.

Finding the Angle Measures

The sine value of −22-\frac{\sqrt{2}}{2} corresponds to the angle measures of 7π4\frac{7\pi}{4} and 11π4\frac{11\pi}{4}. These angle measures are in the third and fourth quadrants, respectively.

Solving the Equation Using the Unit Circle

To solve the equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2} using the unit circle, we need to find the angle measures that correspond to the sine value of −22-\frac{\sqrt{2}}{2}. We can use the following steps:

  1. Draw the unit circle and mark the point that corresponds to the sine value of −22-\frac{\sqrt{2}}{2}.
  2. Find the angle measures that correspond to the point on the unit circle.
  3. Write the angle measures in radians.

Step 1: Drawing the Unit Circle

To draw the unit circle, we need to draw a circle with a radius of 1, centered at the origin of the coordinate plane.

Step 2: Marking the Point

To mark the point that corresponds to the sine value of −22-\frac{\sqrt{2}}{2}, we need to find the point on the unit circle that has a sine value of −22-\frac{\sqrt{2}}{2}. We can use the following steps:

  1. Draw a line from the origin to the point on the unit circle.
  2. Measure the angle between the line and the x-axis.
  3. Write the angle measure in radians.

Step 3: Finding the Angle Measures

To find the angle measures that correspond to the point on the unit circle, we need to use the following steps:

  1. Find the angle measures that correspond to the sine value of −22-\frac{\sqrt{2}}{2}.
  2. Write the angle measures in radians.

Solving the Equation Using Trigonometric Identities

We can also solve the equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2} using trigonometric identities. We can use the following steps:

  1. Use the identity sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta) to rewrite the equation.
  2. Solve the equation using the identity.

Step 1: Rewriting the Equation

To rewrite the equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2} using the identity sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta), we need to use the following steps:

  1. Rewrite the equation as sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta).
  2. Simplify the equation.

Step 2: Solving the Equation

To solve the equation sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta), we need to use the following steps:

  1. Use the identity sin⁡(π−θ)=sin⁡θ\sin (\pi - \theta) = \sin \theta to rewrite the equation.
  2. Solve the equation using the identity.

Conclusion

In this article, we have explored the different methods of solving the equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2}. We have used the unit circle to visualize the solution and have also used trigonometric identities to solve the equation. We have found that the angle measures that correspond to the sine value of −22-\frac{\sqrt{2}}{2} are 7π4\frac{7\pi}{4} and 11π4\frac{11\pi}{4}.

Final Answer

Q: What is the sine function?

A: The sine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.

Q: How do I use the unit circle to solve trigonometric equations?

A: To use the unit circle to solve trigonometric equations, you need to visualize the sine and cosine functions as the ratios of the coordinates of a point on the unit circle to the radius of the circle. You can then use the unit circle to find the angle measures that correspond to the sine value of the equation.

Q: What is the period of the sine function?

A: The period of the sine function is 2π2\pi, which means that it repeats every 2π2\pi radians.

Q: How do I rewrite the equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2} using the identity sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta)?

A: To rewrite the equation sin⁡θ=−22\sin \theta = -\frac{\sqrt{2}}{2} using the identity sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta), you need to use the following steps:

  1. Rewrite the equation as sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta).
  2. Simplify the equation.

Q: How do I solve the equation sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta)?

A: To solve the equation sin⁡θ=−sin⁡(π−θ)\sin \theta = -\sin (\pi - \theta), you need to use the following steps:

  1. Use the identity sin⁡(π−θ)=sin⁡θ\sin (\pi - \theta) = \sin \theta to rewrite the equation.
  2. Solve the equation using the identity.

Q: What are the angle measures that correspond to the sine value of −22-\frac{\sqrt{2}}{2}?

A: The angle measures that correspond to the sine value of −22-\frac{\sqrt{2}}{2} are 7π4\frac{7\pi}{4} and 11π4\frac{11\pi}{4}.

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

A: To use trigonometric identities to solve trigonometric equations, you need to use the following steps:

  1. Identify the trigonometric identity that can be used to rewrite the equation.
  2. Rewrite the equation using the trigonometric identity.
  3. Solve the equation using the trigonometric identity.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin⁡(π−θ)=sin⁡θ\sin (\pi - \theta) = \sin \theta
  • cos⁡(π−θ)=−cos⁡θ\cos (\pi - \theta) = -\cos \theta
  • sin⁡(π+θ)=−sin⁡θ\sin (\pi + \theta) = -\sin \theta
  • cos⁡(π+θ)=−cos⁡θ\cos (\pi + \theta) = -\cos \theta

Q: How do I use the unit circle to find the angle measures that correspond to the sine value of an equation?

A: To use the unit circle to find the angle measures that correspond to the sine value of an equation, you need to visualize the sine and cosine functions as the ratios of the coordinates of a point on the unit circle to the radius of the circle. You can then use the unit circle to find the angle measures that correspond to the sine value of the equation.

Q: What are some tips for solving trigonometric equations?

A: Some tips for solving trigonometric equations include:

  • Use the unit circle to visualize the sine and cosine functions.
  • Use trigonometric identities to rewrite the equation.
  • Solve the equation using the trigonometric identity.
  • Check your answer by plugging it back into the original equation.

Conclusion

In this article, we have answered some frequently asked questions about solving trigonometric equations. We have covered topics such as the sine function, the unit circle, trigonometric identities, and tips for solving trigonometric equations. We hope that this article has been helpful in answering your questions and providing you with a better understanding of trigonometric equations.