Explain How To Find The Exact Value Of \[$\sin(570^\circ)\$\].

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Introduction

In trigonometry, finding the exact value of a sine function can be a challenging task, especially when dealing with angles in degrees. The sine function is periodic with a period of $360^\circ$, meaning that the value of $\sin(x)$ repeats every $360^\circ$. However, when dealing with angles greater than $360^\circ$, we need to use the periodic property of the sine function to find the exact value. In this article, we will explain how to find the exact value of $\sin(570^\circ)$.

The Periodic Property of the Sine Function

The sine function has a periodic property, which states that $\sin(x) = \sin(x + 360^\circ)$ for any angle $x$. This means that if we add or subtract multiples of $360^\circ$ to an angle, the value of the sine function remains the same.

Reducing the Angle to a Smaller Range

To find the exact value of $\sin(570^\circ)$, we need to reduce the angle to a smaller range, typically between $0^\circ$ and $360^\circ$. We can do this by subtracting multiples of $360^\circ$ from the angle until we get a value within this range.

Step 1: Subtract $360^\circ$ from the Angle

Let's start by subtracting $360^\circ$ from $570^\circ$:

$$570^\circ - 360^\circ = 210^\circ$$

Step 2: Find the Exact Value of $\sin(210^\circ)$

Now that we have reduced the angle to $210^\circ$, we need to find the exact value of $\sin(210^\circ)$. We can use the unit circle or trigonometric identities to find this value.

Using the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine function is defined as the y-coordinate of a point on the unit circle. We can use the unit circle to find the exact value of $\sin(210^\circ)$.

Step 3: Find the Coordinates of the Point on the Unit Circle

To find the coordinates of the point on the unit circle corresponding to an angle of $210^\circ$, we need to use the following formulas:

$$x = \cos(\theta)$$

$$y = \sin(\theta)$$

where $\theta$ is the angle in radians.

Converting the Angle to Radians

To convert the angle from degrees to radians, we can use the following formula:

$$\theta = \frac{\pi}{180} \times \theta_{\text{degrees}}$$

where $\theta_{\text{degrees}}$ is the angle in degrees.

Step 4: Find the Exact Value of $\sin(210^\circ)$

Now that we have the coordinates of the point on the unit circle corresponding to an angle of $210^\circ$, we can find the exact value of $\sin(210^\circ)$.

Using Trigonometric Identities

We can also use trigonometric identities to find the exact value of $\sin(210^\circ)$. One such identity is:

$$\sin(210^\circ) = \sin(180^\circ + 30^\circ)$$

Using this identity, we can find the exact value of $\sin(210^\circ)$.

Finding the Exact Value of $\sin(210^\circ)$

Using the unit circle or trigonometric identities, we can find the exact value of $\sin(210^\circ)$:

$$\sin(210^\circ) = -\frac{\sqrt{3}}{2}$$

Conclusion

In this article, we explained how to find the exact value of $\sin(570^\circ)$. We used the periodic property of the sine function to reduce the angle to a smaller range, and then used the unit circle or trigonometric identities to find the exact value of $\sin(210^\circ)$. The exact value of $\sin(570^\circ)$ is:

$$\sin(570^\circ) = -\frac{\sqrt{3}}{2}$$

Final Answer

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Q: What is the periodic property of the sine function?

A: The periodic property of the sine function states that $\sin(x) = \sin(x + 360^\circ)$ for any angle $x$. This means that if we add or subtract multiples of $360^\circ$ to an angle, the value of the sine function remains the same.

Q: How do I reduce an angle to a smaller range?

A: To reduce an angle to a smaller range, typically between $0^\circ$ and $360^\circ$, we can subtract multiples of $360^\circ$ from the angle until we get a value within this range.

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 sine function is defined as the y-coordinate of a point on the unit circle.

Q: How do I find the coordinates of a point on the unit circle?

A: To find the coordinates of a point on the unit circle corresponding to an angle $\theta$, we can use the following formulas:

$$x = \cos(\theta)$$

$$y = \sin(\theta)$$

Q: How do I convert an angle from degrees to radians?

A: To convert an angle from degrees to radians, we can use the following formula:

$$\theta = \frac{\pi}{180} \times \theta_{\text{degrees}}$$

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin(180^\circ + \theta) = -\sin(\theta)$
• $\cos(180^\circ + \theta) = -\cos(\theta)$
• $\sin(360^\circ - \theta) = \sin(\theta)$
• $\cos(360^\circ - \theta) = \cos(\theta)$

Q: How do I find the exact value of $\sin(570^\circ)$?

A: To find the exact value of $\sin(570^\circ)$, we can use the periodic property of the sine function to reduce the angle to a smaller range, and then use the unit circle or trigonometric identities to find the exact value.

Q: What is the final answer for $\sin(570^\circ)$?

A: The final answer for $\sin(570^\circ)$ is:

$$\sin(570^\circ) = -\frac{\sqrt{3}}{2}$$

Q: Can I use a calculator to find the exact value of $\sin(570^\circ)$?

A: Yes, you can use a calculator to find the exact value of $\sin(570^\circ)$. However, keep in mind that calculators may not always give you the exact value, especially for angles that are not in the standard range of $0^\circ$ to $360^\circ$.

Q: What are some common mistakes to avoid when finding the exact value of $\sin(570^\circ)$?

A: Some common mistakes to avoid when finding the exact value of $\sin(570^\circ)$ include:

• Not using the periodic property of the sine function to reduce the angle to a smaller range
• Not using the unit circle or trigonometric identities to find the exact value
• Not converting the angle from degrees to radians
• Not using the correct trigonometric identities

Q: How can I practice finding the exact value of $\sin(570^\circ)$?

A: You can practice finding the exact value of $\sin(570^\circ)$ by working through examples and exercises in your textbook or online resources. You can also try using different angles and trigonometric functions to practice your skills.