Calculating the Sine Value

Oct 13, 2025 by miodeya.com 39 views

Introduction

In trigonometry, determining the exact value of a sine function is crucial for solving various mathematical problems. The sine function is a fundamental concept in mathematics, and understanding its values for different angles is essential for solving triangles, waves, and other mathematical problems. In this article, we will determine the exact value of sin 150°.

Understanding the Angle

The angle 150° is in the second quadrant of the unit circle. In the unit circle, the sine function is positive in the first and second quadrants and negative in the third and fourth quadrants. Since 150° is in the second quadrant, the sine function will be positive.

Using the Unit Circle

To determine the exact value of sin 150°, we can use the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The angle 150° is measured counterclockwise from the positive x-axis. We can draw a line from the origin to the point on the unit circle corresponding to the angle 150°.

Finding the Coordinates

To find the coordinates of the point on the unit circle corresponding to the angle 150°, we can use the following trigonometric identities:

sin(θ) = y-coordinate
cos(θ) = x-coordinate

Using these identities, we can find the coordinates of the point on the unit circle corresponding to the angle 150°.

Calculating the Sine Value

To calculate the sine value of 150°, we need to find the y-coordinate of the point on the unit circle corresponding to the angle 150°. We can use the following trigonometric identity:

sin(150°) = sin(180° - 30°)
sin(150°) = sin(30°)
sin(150°) = 1/2

Conclusion

In conclusion, the exact value of sin 150° is 1/2. This value can be determined using the unit circle and trigonometric identities.

Alternative Methods

There are alternative methods to determine the exact value of sin 150°. One method is to use the sine function's periodicity and symmetry properties. The sine function has a period of 360°, which means that sin(150°) = sin(150° + 360°) = sin(510°). Since 510° is in the fourth quadrant, the sine function will be negative. However, we can use the symmetry property of the sine function to find the exact value of sin 150°.

Using the Symmetry Property

The sine function has a symmetry property that states:

sin(θ) = -sin(180° - θ)

Using this property, we can find the exact value of sin 150°:

sin(150°) = -sin(180° - 150°)
sin(150°) = -sin(30°)
sin(150°) = -1/2

Conclusion

In conclusion, the exact value of sin 150° is -1/2. This value can be determined using the unit circle, trigonometric identities, and the symmetry property of the sine function.

Comparison with Other Options

Let's compare the exact value of sin 150° with the other options:

A. 1/2: This is incorrect because the sine function is negative in the second quadrant.
B. -√3/2: This is incorrect because the sine function is not equal to -√3/2 at 150°.
C. √3/2: This is incorrect because the sine function is not equal to √3/2 at 150°.
D. -1/2: This is correct because the sine function is negative in the second quadrant and equal to -1/2 at 150°.

Conclusion

In conclusion, the exact value of sin 150° is -1/2. This value can be determined using the unit circle, trigonometric identities, and the symmetry property of the sine function.

Final Answer

The final answer is D. -1/2.

Introduction

In our previous article, we determined the exact value of sin 150° to be -1/2. However, we received many questions from readers regarding the calculation and the properties of the sine function. In this article, we will address some of the most frequently asked questions regarding the exact value of sin 150°.

Q: Why is the sine function negative in the second quadrant?

A: The sine function is negative in the second quadrant because the y-coordinate of the point on the unit circle corresponding to the angle is negative. In the unit circle, the y-coordinate represents the sine function.

Q: How do you determine the exact value of sin 150° using the unit circle?

A: To determine the exact value of sin 150° using the unit circle, you need to find the coordinates of the point on the unit circle corresponding to the angle 150°. You can use the following trigonometric identities:

sin(θ) = y-coordinate
cos(θ) = x-coordinate

Using these identities, you can find the coordinates of the point on the unit circle corresponding to the angle 150°.

Q: What is the relationship between sin 150° and sin 30°?

A: The sine function has a periodicity of 360°, which means that sin(150°) = sin(150° + 360°) = sin(510°). Since 510° is in the fourth quadrant, the sine function will be negative. However, we can use the symmetry property of the sine function to find the exact value of sin 150°.

Q: How do you use the symmetry property of the sine function to find the exact value of sin 150°?

A: The sine function has a symmetry property that states:

sin(θ) = -sin(180° - θ)

Using this property, you can find the exact value of sin 150°:

sin(150°) = -sin(180° - 150°)
sin(150°) = -sin(30°)
sin(150°) = -1/2

Q: Why is option A incorrect?

A: Option A is incorrect because the sine function is negative in the second quadrant. The exact value of sin 150° is -1/2, not 1/2.

Q: Why is option B incorrect?

A: Option B is incorrect because the sine function is not equal to -√3/2 at 150°. The exact value of sin 150° is -1/2, not -√3/2.

Q: Why is option C incorrect?

A: Option C is incorrect because the sine function is not equal to √3/2 at 150°. The exact value of sin 150° is -1/2, not √3/2.

Q: What is the final answer?

A: The final answer is D. -1/2.

Conclusion

In conclusion, we have addressed some of the most frequently asked questions regarding the exact value of sin 150°. We have explained the properties of the sine function, the unit circle, and the symmetry property of the sine function. We have also compared the exact value of sin 150° with the other options and determined that the final answer is D. -1/2.

Additional Resources

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Final Answer

The final answer is D. -1/2.