Determine The Angle Between ${ 0\$} And ${ 2\pi\$} That Is Coterminal To ${ 780^{\circ}\$} .A. { \frac{10\pi}{3}$}$ B. { \frac{5\pi}{3}$}$ C. { \frac{4\pi}{3}$}$ D. { \frac{\pi}{3}$}$

Determine the Angle Between $0$ and $2\pi$ That is Coterminal to $780^{\circ}$

In trigonometry, angles are measured in degrees or radians, and understanding the relationship between these measurements is crucial for solving various problems. One concept that is essential in this context is coterminal angles. Coterminal angles are angles that have the same terminal side when drawn in standard position. In this article, we will explore how to determine the angle between $0$ and $2\pi$ that is coterminal to $780^{\circ}$.

Coterminal angles are angles that have the same terminal side when drawn in standard position. This means that if two angles are coterminal, they will have the same sine, cosine, and tangent values. To find the coterminal angle of a given angle, we can add or subtract multiples of $360^{\circ}$ (or $2\pi$ radians) from the given angle.

To find the coterminal angle of $780^{\circ}$, we need to subtract multiples of $360^{\circ}$ from $780^{\circ}$ until we get an angle between $0^{\circ}$ and $360^{\circ}$. Let's do this step by step:

1. $780^{\circ} - 360^{\circ} = 420^{\circ}$
2. $420^{\circ} - 360^{\circ} = 60^{\circ}$

Now, we have an angle between $0^{\circ}$ and $360^{\circ}$, which is $60^{\circ}$. However, we need to find the angle between $0$ and $2\pi$ that is coterminal to $780^{\circ}$. To do this, we can convert the angle from degrees to radians.

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

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Using this formula, we can convert $60^{\circ}$ to radians:

$$\frac{60^{\circ} \times \pi}{180} = \frac{\pi}{3}$$

In conclusion, the angle between $0$ and $2\pi$ that is coterminal to $780^{\circ}$ is $\frac{\pi}{3}$. This is because we can add or subtract multiples of $2\pi$ from $\frac{\pi}{3}$ to get an angle that is coterminal to $780^{\circ}$.

The correct answer is:

• D. $\frac{\pi}{3}$

Here are a few more examples of finding coterminal angles:

• Find the coterminal angle of $900^{\circ}$.
• Find the coterminal angle of $540^{\circ}$.
• Find the coterminal angle of $720^{\circ}$.

• To find the coterminal angle of $900^{\circ}$, we can subtract multiples of $360^{\circ}$ from $900^{\circ}$ until we get an angle between $0^{\circ}$ and $360^{\circ}$. Let's do this step by step:

1. $900^{\circ} - 360^{\circ} = 540^{\circ}$
2. $540^{\circ} - 360^{\circ} = 180^{\circ}$

Now, we have an angle between $0^{\circ}$ and $360^{\circ}$, which is $180^{\circ}$. To convert this angle from degrees to radians, we can use the following formula:

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Using this formula, we can convert $180^{\circ}$ to radians:

$$\frac{180^{\circ} \times \pi}{180} = \pi$$

Therefore, the coterminal angle of $900^{\circ}$ is $\pi$.

• To find the coterminal angle of $540^{\circ}$, we can subtract multiples of $360^{\circ}$ from $540^{\circ}$ until we get an angle between $0^{\circ}$ and $360^{\circ}$. Let's do this step by step:

1. $540^{\circ} - 360^{\circ} = 180^{\circ}$

Now, we have an angle between $0^{\circ}$ and $360^{\circ}$, which is $180^{\circ}$. To convert this angle from degrees to radians, we can use the following formula:

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Using this formula, we can convert $180^{\circ}$ to radians:

$$\frac{180^{\circ} \times \pi}{180} = \pi$$

Therefore, the coterminal angle of $540^{\circ}$ is $\pi$.

• To find the coterminal angle of $720^{\circ}$, we can subtract multiples of $360^{\circ}$ from $720^{\circ}$ until we get an angle between $0^{\circ}$ and $360^{\circ}$. Let's do this step by step:

1. $720^{\circ} - 360^{\circ} = 360^{\circ}$

Now, we have an angle between $0^{\circ}$ and $360^{\circ}$, which is $360^{\circ}$. To convert this angle from degrees to radians, we can use the following formula:

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Using this formula, we can convert $360^{\circ}$ to radians:

$$\frac{360^{\circ} \times \pi}{180} = 2\pi$$

$$$$

Q: What is a coterminal angle?

A: A coterminal angle is an angle that has the same terminal side when drawn in standard position. This means that if two angles are coterminal, they will have the same sine, cosine, and tangent values.

Q: How do I find the coterminal angle of a given angle?

A: To find the coterminal angle of a given angle, you can add or subtract multiples of $360^{\circ}$ (or $2\pi$ radians) from the given angle. For example, if you want to find the coterminal angle of $780^{\circ}$, you can subtract $360^{\circ}$ from $780^{\circ}$ to get $420^{\circ}$, and then subtract $360^{\circ}$ from $420^{\circ}$ to get $60^{\circ}$.

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

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

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

For example, if you want to convert $60^{\circ}$ to radians, you can use the following formula:

$$\frac{60^{\circ} \times \pi}{180} = \frac{\pi}{3}$$

Q: What is the difference between coterminal angles and supplementary angles?

A: Coterminal angles are angles that have the same terminal side when drawn in standard position, while supplementary angles are angles that add up to $180^{\circ}$ (or $\pi$ radians). For example, $60^{\circ}$ and $120^{\circ}$ are supplementary angles, but they are not coterminal angles.

Q: Can you give me some examples of finding coterminal angles?

A: Here are a few examples:

• Find the coterminal angle of $900^{\circ}$.
• Find the coterminal angle of $540^{\circ}$.
• Find the coterminal angle of $720^{\circ}$.

A: Solutions to Examples

• To find the coterminal angle of $900^{\circ}$, we can subtract multiples of $360^{\circ}$ from $900^{\circ}$ until we get an angle between $0^{\circ}$ and $360^{\circ}$. Let's do this step by step:

1. $900^{\circ} - 360^{\circ} = 540^{\circ}$
2. $540^{\circ} - 360^{\circ} = 180^{\circ}$

Now, we have an angle between $0^{\circ}$ and $360^{\circ}$, which is $180^{\circ}$. To convert this angle from degrees to radians, we can use the following formula:

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Using this formula, we can convert $180^{\circ}$ to radians:

$$\frac{180^{\circ} \times \pi}{180} = \pi$$

Therefore, the coterminal angle of $900^{\circ}$ is $\pi$.

• To find the coterminal angle of $540^{\circ}$, we can subtract multiples of $360^{\circ}$ from $540^{\circ}$ until we get an angle between $0^{\circ}$ and $360^{\circ}$. Let's do this step by step:

1. $540^{\circ} - 360^{\circ} = 180^{\circ}$

Now, we have an angle between $0^{\circ}$ and $360^{\circ}$, which is $180^{\circ}$. To convert this angle from degrees to radians, we can use the following formula:

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Using this formula, we can convert $180^{\circ}$ to radians:

$$\frac{180^{\circ} \times \pi}{180} = \pi$$

Therefore, the coterminal angle of $540^{\circ}$ is $\pi$.

• To find the coterminal angle of $720^{\circ}$, we can subtract multiples of $360^{\circ}$ from $720^{\circ}$ until we get an angle between $0^{\circ}$ and $360^{\circ}$. Let's do this step by step:

1. $720^{\circ} - 360^{\circ} = 360^{\circ}$

Now, we have an angle between $0^{\circ}$ and $360^{\circ}$, which is $360^{\circ}$. To convert this angle from degrees to radians, we can use the following formula:

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Using this formula, we can convert $360^{\circ}$ to radians:

$$\frac{360^{\circ} \times \pi}{180} = 2\pi$$

Therefore, the coterminal angle of $720^{\circ}$ is $2\pi$.

In conclusion, coterminal angles are angles that have the same terminal side when drawn in standard position. To find the coterminal angle of a given angle, you can add or subtract multiples of $360^{\circ}$ (or $2\pi$ radians) from the given angle. We have also discussed how to convert an angle from degrees to radians and provided examples of finding coterminal angles.