Find The Exact Values:${ \begin{array}{l} \sin \frac{2 \pi}{3}= \ \cos \frac{2 \pi}{3}= \ \tan \frac{2 \pi}{3}= \end{array} }$

Introduction

Trigonometric functions are a crucial part of mathematics, and understanding their values is essential for solving various mathematical problems. In this article, we will focus on finding the exact values of sine, cosine, and tangent functions for a specific angle, which is $\frac{2 \pi}{3}$ radians. We will use the unit circle and trigonometric identities to derive these values.

The Unit Circle

The unit circle is a fundamental concept in trigonometry, and it is used to define the trigonometric functions. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The angle $\theta$ is measured counterclockwise from the positive x-axis. The coordinates of a point on the unit circle are given by $(\cos \theta, \sin \theta)$.

Finding the Exact Values

Sine of $\frac{2 \pi}{3}$

To find the exact value of $\sin \frac{2 \pi}{3}$, we can use the unit circle. We know that the angle $\frac{2 \pi}{3}$ is in the second quadrant, where the sine function is positive. We can draw a right triangle in the second quadrant with an angle of $\frac{2 \pi}{3}$ and a hypotenuse of length 1. The opposite side of the angle has a length of $\frac{\sqrt{3}}{2}$, and the adjacent side has a length of $\frac{1}{2}$. Therefore, we can write:

$$\sin \frac{2 \pi}{3} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2}$$

Cosine of $\frac{2 \pi}{3}$

To find the exact value of $\cos \frac{2 \pi}{3}$, we can use the unit circle. We know that the angle $\frac{2 \pi}{3}$ is in the second quadrant, where the cosine function is negative. We can draw a right triangle in the second quadrant with an angle of $\frac{2 \pi}{3}$ and a hypotenuse of length 1. The opposite side of the angle has a length of $\frac{\sqrt{3}}{2}$, and the adjacent side has a length of $-\frac{1}{2}$. Therefore, we can write:

$$\cos \frac{2 \pi}{3} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-\frac{1}{2}}{1} = -\frac{1}{2}$$

Tangent of $\frac{2 \pi}{3}$

To find the exact value of $\tan \frac{2 \pi}{3}$, we can use the definition of tangent as the ratio of sine and cosine:

$$\tan \frac{2 \pi}{3} = \frac{\sin \frac{2 \pi}{3}}{\cos \frac{2 \pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}$$

Conclusion

In this article, we have found the exact values of sine, cosine, and tangent functions for the angle $\frac{2 \pi}{3}$ radians. We used the unit circle and trigonometric identities to derive these values. The exact values are:

• $\sin \frac{2 \pi}{3} = \frac{\sqrt{3}}{2}$
• $\cos \frac{2 \pi}{3} = -\frac{1}{2}$
• $\tan \frac{2 \pi}{3} = -\sqrt{3}$

These values are essential for solving various mathematical problems, and they can be used to derive other trigonometric identities.

Applications

The exact values of sine, cosine, and tangent functions have numerous applications in mathematics and other fields. Some of the applications include:

• Trigonometric identities: The exact values of sine, cosine, and tangent functions can be used to derive other trigonometric identities, such as the Pythagorean identity.
• Solving triangles: The exact values of sine, cosine, and tangent functions can be used to solve triangles, which is essential in geometry and trigonometry.
• Waves and vibrations: The exact values of sine, cosine, and tangent functions can be used to model waves and vibrations, which is essential in physics and engineering.
• Computer graphics: The exact values of sine, cosine, and tangent functions can be used in computer graphics to create 3D models and animations.

Final Thoughts

In conclusion, finding the exact values of sine, cosine, and tangent functions is an essential part of mathematics. The unit circle and trigonometric identities are used to derive these values, and they have numerous applications in mathematics and other fields. The exact values of sine, cosine, and tangent functions are:

• $\sin \frac{2 \pi}{3} = \frac{\sqrt{3}}{2}$
• $\cos \frac{2 \pi}{3} = -\frac{1}{2}$
• $\tan \frac{2 \pi}{3} = -\sqrt{3}$

Introduction

In our previous article, we discussed finding the exact values of sine, cosine, and tangent functions for a specific angle, which is $\frac{2 \pi}{3}$ radians. In this article, we will answer some frequently asked questions related to trigonometric functions.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The angle $\theta$ is measured counterclockwise from the positive x-axis. The coordinates of a point on the unit circle are given by $(\cos \theta, \sin \theta)$.

Q: How do I find the exact value of a trigonometric function?

A: To find the exact value of a trigonometric function, you can use the unit circle and trigonometric identities. For example, to find the exact value of $\sin \frac{2 \pi}{3}$, you can draw a right triangle in the second quadrant with an angle of $\frac{2 \pi}{3}$ and a hypotenuse of length 1. The opposite side of the angle has a length of $\frac{\sqrt{3}}{2}$, and the adjacent side has a length of $\frac{1}{2}$. Therefore, you can write:

$$\sin \frac{2 \pi}{3} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\frac{\sqrt{3}}{2}}{1} = \frac{\sqrt{3}}{2}$$

Q: What is the difference between sine and cosine?

A: Sine and cosine are two trigonometric functions that are defined in terms of the coordinates of a point on the unit circle. The sine function is defined as the ratio of the opposite side to the hypotenuse, while the cosine function is defined as the ratio of the adjacent side to the hypotenuse.

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

A: Trigonometric identities are equations that relate different trigonometric functions. For example, the Pythagorean identity states that:

$$\sin^2 \theta + \cos^2 \theta = 1$$

You can use this identity to solve problems by substituting the values of sine and cosine into the equation.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
• Quotient identity: $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• Reciprocal identity: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$

Q: How do I use trigonometric functions to solve triangles?

A: To solve a triangle, you can use trigonometric functions to find the lengths of the sides and the measures of the angles. For example, if you know the length of one side and the measure of one angle, you can use the sine function to find the length of the adjacent side.

Q: What are some real-world applications of trigonometric functions?

A: Trigonometric functions have numerous real-world applications, including:

• Navigation: Trigonometric functions are used in navigation to determine the position and direction of a ship or aircraft.
• Physics: Trigonometric functions are used in physics to describe the motion of objects and the behavior of waves.
• Engineering: Trigonometric functions are used in engineering to design and analyze systems, such as bridges and buildings.
• Computer graphics: Trigonometric functions are used in computer graphics to create 3D models and animations.

Conclusion

In this article, we have answered some frequently asked questions related to trigonometric functions. We have discussed the unit circle, trigonometric identities, and real-world applications of trigonometric functions. We hope that this article has been helpful in understanding trigonometric functions and their applications.