Radian Measures and the Unit Circle

Oct 13, 2025 by windailys.com 30 views

Introduction to the Unit Circle

The unit circle is a fundamental concept in mathematics, particularly in trigonometry and geometry. It is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The unit circle is used to define the trigonometric functions of sine, cosine, and tangent, and it plays a crucial role in solving problems involving right triangles and circular functions.

Radian Measures and the Unit Circle

A radian measure is a way of measuring angles in terms of the radius of a circle. It is defined as the ratio of the arc length to the radius of the circle. In other words, if an arc subtends an angle of $\theta$ radians at the center of a circle, then the length of the arc is equal to $\theta$ times the radius of the circle.

Finding the Corresponding Point on the Unit Circle

To find the corresponding point on the unit circle for a given radian measure $\theta$ , we need to use the trigonometric functions of sine and cosine. The point on the unit circle corresponding to an angle $\theta$ is given by the coordinates $(\cos \theta, \sin \theta)$ .

Calculating the Coordinates

In this case, we are given the radian measure $\theta = \frac{7 \pi}{6}$ . To find the corresponding point on the unit circle, we need to calculate the values of $\cos \frac{7 \pi}{6}$ and $\sin \frac{7 \pi}{6}$ .

Using Trigonometric Identities

We can use the trigonometric identities for cosine and sine of sum and difference of angles to simplify the calculations. Specifically, we can use the identity $\cos (A + B) = \cos A \cos B - \sin A \sin B$ and $\sin (A + B) = \sin A \cos B + \cos A \sin B$ .

Applying the Identities

Using the identity $\cos (A + B) = \cos A \cos B - \sin A \sin B$ , we can write $\cos \frac{7 \pi}{6} = \cos \left(\frac{\pi}{6} + \pi\right) = \cos \frac{\pi}{6} \cos \pi - \sin \frac{\pi}{6} \sin \pi$ . Similarly, using the identity $\sin (A + B) = \sin A \cos B + \cos A \sin B$ , we can write $\sin \frac{7 \pi}{6} = \sin \left(\frac{\pi}{6} + \pi\right) = \sin \frac{\pi}{6} \cos \pi + \cos \frac{\pi}{6} \sin \pi$ .

Evaluating the Trigonometric Functions

We know that $\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$ , $\sin \frac{\pi}{6} = \frac{1}{2}$ , $\cos \pi = -1$ , and $\sin \pi = 0$ . Substituting these values into the expressions for $\cos \frac{7 \pi}{6}$ and $\sin \frac{7 \pi}{6}$ , we get $\cos \frac{7 \pi}{6} = \frac{\sqrt{3}}{2} (-1) - \frac{1}{2} (0) = -\frac{\sqrt{3}}{2}$ and $\sin \frac{7 \pi}{6} = \frac{1}{2} (-1) + \frac{\sqrt{3}}{2} (0) = -\frac{1}{2}$ .

Conclusion

Therefore, the corresponding point on the unit circle for the given radian measure $\theta = \frac{7 \pi}{6}$ is $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ . This point lies in the third quadrant of the unit circle, where the x-coordinate is negative and the y-coordinate is negative.

Comparison with Other Options

Let's compare the calculated point with the other options given in the problem. Option A is $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ , which matches our calculated point. Option B is $\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)$ , which does not match our calculated point.

Conclusion

In conclusion, the correct answer is option A, $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ . This point corresponds to the given radian measure $\theta = \frac{7 \pi}{6}$ on the unit circle.

Final Answer

The final answer is: $\boxed{\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)}$

Introduction

The unit circle and radian measures are fundamental concepts in mathematics, particularly in trigonometry and geometry. In this article, we will provide a comprehensive Q&A guide to help you understand the unit circle and radian measures.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is used to define the trigonometric functions of sine, cosine, and tangent.

Q: What is a radian measure?

A: A radian measure is a way of measuring angles in terms of the radius of a circle. It is defined as the ratio of the arc length to the radius of the circle.

Q: How do I find the corresponding point on the unit circle for a given radian measure?

A: To find the corresponding point on the unit circle for a given radian measure, you need to use the trigonometric functions of sine and cosine. The point on the unit circle corresponding to an angle $\theta$ is given by the coordinates $(\cos \theta, \sin \theta)$ .

Q: What are the coordinates of the point on the unit circle corresponding to the radian measure $\theta = \frac{7 \pi}{6}$ ?

A: To find the coordinates of the point on the unit circle corresponding to the radian measure $\theta = \frac{7 \pi}{6}$ , you need to calculate the values of $\cos \frac{7 \pi}{6}$ and $\sin \frac{7 \pi}{6}$ . Using the trigonometric identities for cosine and sine of sum and difference of angles, you can simplify the calculations.

Q: How do I use the trigonometric identities for cosine and sine of sum and difference of angles?

A: You can use the identity $\cos (A + B) = \cos A \cos B - \sin A \sin B$ and $\sin (A + B) = \sin A \cos B + \cos A \sin B$ to simplify the calculations.

Q: What are the values of $\cos \frac{7 \pi}{6}$ and $\sin \frac{7 \pi}{6}$ ?

A: Using the trigonometric identities for cosine and sine of sum and difference of angles, you can write $\cos \frac{7 \pi}{6} = \cos \left(\frac{\pi}{6} + \pi\right) = \cos \frac{\pi}{6} \cos \pi - \sin \frac{\pi}{6} \sin \pi$ and $\sin \frac{7 \pi}{6} = \sin \left(\frac{\pi}{6} + \pi\right) = \sin \frac{\pi}{6} \cos \pi + \cos \frac{\pi}{6} \sin \pi$ . Evaluating the trigonometric functions, you get $\cos \frac{7 \pi}{6} = -\frac{\sqrt{3}}{2}$ and $\sin \frac{7 \pi}{6} = -\frac{1}{2}$ .

Q: What is the corresponding point on the unit circle for the given radian measure $\theta = \frac{7 \pi}{6}$ ?

A: The corresponding point on the unit circle for the given radian measure $\theta = \frac{7 \pi}{6}$ is $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ .

Q: How do I compare the calculated point with the other options given in the problem?

A: You can compare the calculated point with the other options given in the problem by checking if the coordinates match.

Q: What is the correct answer?

A: The correct answer is option A, $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ .

Conclusion

In conclusion, the unit circle and radian measures are fundamental concepts in mathematics, particularly in trigonometry and geometry. By understanding the unit circle and radian measures, you can solve problems involving right triangles and circular functions.

Final Answer

The final answer is: $\boxed{\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)}$