What Is The Value Of \cos \left(150^{\circ}\right ]?A. 3 2 \frac{\sqrt{3}}{2} 2 3 ​ ​ B. − 3 2 -\frac{\sqrt{3}}{2} − 2 3 ​ ​ C. 1 2 \frac{1}{2} 2 1 ​ D. − 1 2 -\frac{1}{2} − 2 1 ​

Introduction

In trigonometry, the cosine function is a fundamental concept that helps us understand the relationships between the angles and side lengths of triangles. The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. In this article, we will explore the value of $\cos \left(150^{\circ}\right)$ and discuss the different methods to calculate it.

Understanding the Cosine Function

The cosine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. The cosine function has a period of $360^{\circ}$, which means that the value of the cosine function repeats every $360^{\circ}$. The cosine function is also an even function, which means that $\cos \left(-x\right) = \cos \left(x\right)$ for all values of $x$.

Calculating the Value of $\cos \left(150^{\circ}\right)$

To calculate the value of $\cos \left(150^{\circ}\right)$, we can use the unit circle. The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. The cosine of an angle is equal to the x-coordinate of the point on the unit circle that corresponds to the angle. To find the value of $\cos \left(150^{\circ}\right)$, we can draw a line from the origin to the point on the unit circle that corresponds to the angle $150^{\circ}$.

Using the Unit Circle to Calculate the Value of $\cos \left(150^{\circ}\right)$

To calculate the value of $\cos \left(150^{\circ}\right)$ using the unit circle, we can follow these steps:

1. Draw a line from the origin to the point on the unit circle that corresponds to the angle $150^{\circ}$.
2. Measure the x-coordinate of the point on the unit circle that corresponds to the angle $150^{\circ}$.
3. The x-coordinate of the point on the unit circle that corresponds to the angle $150^{\circ}$ is equal to the value of $\cos \left(150^{\circ}\right)$.

Using Trigonometric Identities to Calculate the Value of $\cos \left(150^{\circ}\right)$

We can also use trigonometric identities to calculate the value of $\cos \left(150^{\circ}\right)$. One of the most useful trigonometric identities is the cosine of the sum of two angles formula:

$$\cos \left(a + b\right) = \cos \left(a\right) \cos \left(b\right) - \sin \left(a\right) \sin \left(b\right)$$

We can use this formula to calculate the value of $\cos \left(150^{\circ}\right)$ by expressing $150^{\circ}$ as the sum of two angles that we know the cosine and sine values of.

Using the Cosine of the Sum of Two Angles Formula to Calculate the Value of $\cos \left(150^{\circ}\right)$

To calculate the value of $\cos \left(150^{\circ}\right)$ using the cosine of the sum of two angles formula, we can follow these steps:

1. Express $150^{\circ}$ as the sum of two angles that we know the cosine and sine values of.
2. Use the cosine of the sum of two angles formula to calculate the value of $\cos \left(150^{\circ}\right)$.
3. Simplify the expression to find the value of $\cos \left(150^{\circ}\right)$.

Using the Reference Angle to Calculate the Value of $\cos \left(150^{\circ}\right)$

We can also use the reference angle to calculate the value of $\cos \left(150^{\circ}\right)$. The reference angle is the acute angle between the terminal side of the angle and the x-axis. The reference angle for $150^{\circ}$ is $30^{\circ}$.

Using the Reference Angle to Calculate the Value of $\cos \left(150^{\circ}\right)$

To calculate the value of $\cos \left(150^{\circ}\right)$ using the reference angle, we can follow these steps:

1. Find the reference angle for $150^{\circ}$.
2. Use the cosine of the reference angle to calculate the value of $\cos \left(150^{\circ}\right)$.
3. Simplify the expression to find the value of $\cos \left(150^{\circ}\right)$.

Conclusion

In conclusion, we have discussed the value of $\cos \left(150^{\circ}\right)$ and explored different methods to calculate it. We have used the unit circle, trigonometric identities, and the reference angle to calculate the value of $\cos \left(150^{\circ}\right)$. The value of $\cos \left(150^{\circ}\right)$ is $-\frac{\sqrt{3}}{2}$.

Final Answer

The final answer is: $\boxed{-\frac{\sqrt{3}}{2}}$

Introduction

In our previous article, we discussed the value of $\cos \left(150^{\circ}\right)$ and explored different methods to calculate it. In this article, we will answer some of the most frequently asked questions about the value of $\cos \left(150^{\circ}\right)$.

Q: What is the value of $\cos \left(150^{\circ}\right)$?

A: The value of $\cos \left(150^{\circ}\right)$ is $-\frac{\sqrt{3}}{2}$.

Q: How do I calculate the value of $\cos \left(150^{\circ}\right)$?

A: There are several methods to calculate the value of $\cos \left(150^{\circ}\right)$. You can use the unit circle, trigonometric identities, or the reference angle to calculate the value of $\cos \left(150^{\circ}\right)$.

Q: What is the unit circle?

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

Q: How do I use the unit circle to calculate the value of $\cos \left(150^{\circ}\right)$?

A: To use the unit circle to calculate the value of $\cos \left(150^{\circ}\right)$, you can draw a line from the origin to the point on the unit circle that corresponds to the angle $150^{\circ}$. Then, measure the x-coordinate of the point on the unit circle that corresponds to the angle $150^{\circ}$. The x-coordinate of the point on the unit circle that corresponds to the angle $150^{\circ}$ is equal to the value of $\cos \left(150^{\circ}\right)$.

Q: What is the reference angle?

A: The reference angle is the acute angle between the terminal side of the angle and the x-axis. The reference angle for $150^{\circ}$ is $30^{\circ}$.

Q: How do I use the reference angle to calculate the value of $\cos \left(150^{\circ}\right)$?

A: To use the reference angle to calculate the value of $\cos \left(150^{\circ}\right)$, you can find the reference angle for $150^{\circ}$, which is $30^{\circ}$. Then, use the cosine of the reference angle to calculate the value of $\cos \left(150^{\circ}\right)$. The cosine of $30^{\circ}$ is $\frac{\sqrt{3}}{2}$, so the value of $\cos \left(150^{\circ}\right)$ is $-\frac{\sqrt{3}}{2}$.

Q: What are some common mistakes to avoid when calculating the value of $\cos \left(150^{\circ}\right)$?

A: Some common mistakes to avoid when calculating the value of $\cos \left(150^{\circ}\right)$ include:

• Using the wrong reference angle
• Not using the correct trigonometric identity
• Not simplifying the expression correctly
• Not checking the units of the answer

Q: How do I check my answer for the value of $\cos \left(150^{\circ}\right)$?

A: To check your answer for the value of $\cos \left(150^{\circ}\right)$, you can use a calculator or a trigonometric table to find the value of $\cos \left(150^{\circ}\right)$. You can also use the unit circle or the reference angle to calculate the value of $\cos \left(150^{\circ}\right)$ and compare it to your answer.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about the value of $\cos \left(150^{\circ}\right)$. We have discussed the different methods to calculate the value of $\cos \left(150^{\circ}\right)$, including the unit circle, trigonometric identities, and the reference angle. We have also discussed some common mistakes to avoid when calculating the value of $\cos \left(150^{\circ}\right)$ and how to check your answer.

Final Answer

The final answer is: $\boxed{-\frac{\sqrt{3}}{2}}$