Question

9
which of the following is the general formula that is coterminal with the angle below?
(1 point)
θ = -119°
○ θ = 119 + 360° (n)
○ θ = -119 + 360° (n)
○ θ = 241 + 360° (n)
○ θ = -241 + 360° (n)

Explanation:

Step1: Recall coterminal angle formula

Coterminal angles are found by adding or subtracting multiples of \(360^\circ\) (for degree measure). The general formula for coterminal angles with an angle \(\theta\) is \(\theta + 360^\circ n\), where \(n\) is any integer.

Step2: Apply formula to given angle

The given angle is \(\theta=- 119^\circ\). So the general formula for its coterminal angles should be \(\theta=-119^\circ + 360^\circ n\), where \(n\in\mathbb{Z}\).

We can also verify by finding a positive coterminal angle: \(-119^\circ+360^\circ = 241^\circ\), so \(\theta = 241^\circ+360^\circ n\) is also correct (since \(-119 + 360n=241+360(n - 1)\), but the first form based on the original angle is \(-119 + 360n\) and the form with \(241\) is also correct. But let's check the options. The option \(\theta=-119 + 360^\circ(n)\) is directly from the formula, and \(\theta = 241+360^\circ(n)\) is also correct (because \(-119+360 = 241\)). Wait, let's check the options again.

Wait, the options are:

1. \(\theta=119 + 360^\circ(n)\)
2. \(\theta=-119 + 360^\circ(n)\)
3. \(\theta=241 + 360^\circ(n)\)
4. \(\theta=-241 + 360^\circ(n)\)

The general formula for coterminal angles with \(\theta=-119^\circ\) is \(\theta=-119^\circ+360^\circ n\) (where \(n\) is integer). Also, if we add \(360^\circ\) to \(-119^\circ\), we get \(241^\circ\), so \(\theta = 241^\circ+360^\circ n\) is also a coterminal angle formula (since \(n\) can be any integer, including \(n\) shifted by 1). But let's check the options. The option \(\theta=-119 + 360^\circ(n)\) is option 2, and \(\theta = 241+360^\circ(n)\) is option 3. Wait, let's calculate:

For \(n = 1\) in \(\theta=-119+360n\), we get \(-119 + 360=241\), so both \(\theta=-119 + 360n\) and \(\theta=241+360n\) are correct. But let's see the question: "the general formula that is coterminal with the angle below". The original angle is \(-119^\circ\), so the general formula starting from \(-119^\circ\) is \(\theta=-119 + 360n\), and the one starting from its positive coterminal angle is \(\theta=241+360n\). Both are correct, but let's check the options.

Wait, let's re - express:

The formula for coterminal angles is \(\text{coterminal angle}=\text{given angle}+360^\circ\times k\), where \(k\in\mathbb{Z}\).

Given angle \(\alpha=-119^\circ\), so coterminal angles are \(-119^\circ+360^\circ k\), \(k\in\mathbb{Z}\).

So the option \(\theta=-119 + 360^\circ(n)\) is correct. Also, when \(k = 1\), \(-119+360 = 241\), so \(\theta=241+360^\circ(n)\) (with \(n=k - 1\)) is also correct. But let's check the options. The second option is \(\theta=-119 + 360^\circ(n)\) and the third is \(\theta=241+360^\circ(n)\). Wait, maybe the question considers the general formula in terms of the original angle. So the correct formula is \(\theta=-119 + 360^\circ(n)\) (option 2) or \(\theta=241+360^\circ(n)\) (option 3). Wait, let's calculate the coterminal angle:

To find a positive coterminal angle with \(-119^\circ\), we add \(360^\circ\): \(-119+360 = 241^\circ\). So the coterminal angles can be written as \(241^\circ+360^\circ n\) (where \(n\) is integer) or \(-119^\circ+360^\circ n\) (since \(241+360n=-119 + 360(n + 1)\)).

Now let's check the options:

- Option 1: \(\theta=119 + 360n\) is wrong, because \(119\) is not related to \(-119\) by coterminal ( \(119-(-119)=238 eq360k\))
- Option 2: \(\theta=-119 + 360n\) is correct (by the formula)
- Option 3: \(\theta=241 + 360n\) is correct (since \(241=-119 + 360\))
- Option 4: \(\theta=-241 + 360n\) is wrong (\(-241-(-119)=-122 eq360k\))

Wait, but the question says "the general formula that is coterminal with the angle below". The angle is \(-119^\circ\), so the general formula is \(\theta=-119^\circ+360^\circ n\) (option 2) and also \(\theta = 241^\circ+360^\circ n\) (option 3). But let's check with \(n = 0\): for option 2, \(\theta=-119^\circ\) (correct, the original angle), for option 3, \(\theta=241^\circ\) (a positive coterminal angle). Both are correct, but maybe the question expects the formula starting with the original angle. So option 2: \(\theta=-119 + 360^\circ(n)\) or option 3: \(\theta=241 + 360^\circ(n)\). Wait, let's check the calculation again.

The formula for coterminal angles in degrees is: if \(\theta\) is an angle, then all coterminal angles are given by \(\theta+360^\circ k\), where \(k\in\mathbb{Z}\). So for \(\theta=-119^\circ\), the coterminal angles are \(-119^\circ+360^\circ k\), \(k\in\mathbb{Z}\). So the option \(\theta=-119 + 360^\circ(n)\) is correct. Also, when \(k = 1\), we get \(241^\circ\), so \(\theta=241^\circ+360^\circ(n - 1)\), which is equivalent to \(\theta=241^\circ+360^\circ n\) (since \(n\) is any integer). So both option 2 and option 3 are correct? Wait, no, let's check the options again.

Wait the options are:

1. \(\theta = 119+360n\)
2. \(\theta=-119 + 360n\)
3. \(\theta=241 + 360n\)
4. \(\theta=-241 + 360n\)

Let's check for \(n = 0\):

- Option 1: \(119 eq - 119\)
- Option 2: \(-119=-119\) (correct, coterminal with itself)
- Option 3: \(241 eq - 119\) (but \(241\) is coterminal with \(-119\) since \(-119 + 360=241\))
- Option 4: \(-241 eq - 119\)

So both option 2 and option 3 are correct? Wait, maybe the question has a typo, but according to the formula, the general formula is \(\theta=-119 + 360n\) (option 2) and \(\theta=241 + 360n\) (option 3). But let's see the problem statement again. It says "the general formula that is coterminal with the angle below". The angle is \(-119^\circ\), so the general formula is \(\theta=-119 + 360n\) (option 2) or \(\theta=241 + 360n\) (option 3). But let's check with \(n = 1\) in option 2: \(-119+360 = 241\), which is the same as option 3 with \(n = 0\). So both are correct, but maybe the intended answer is option 2 or option 3. Wait, let's check the calculation of coterminal angles.

The formula for coterminal angles is: for an angle \(\alpha\), the coterminal angles are \(\alpha+360^\circ k\), \(k\in\mathbb{Z}\). So for \(\alpha=-119^\circ\), the coterminal angles are \(-119^\circ+360^\circ k\), \(k\in\mathbb{Z}\). So option 2 is \(\theta=-119 + 360n\) (where \(n = k\)), which is correct. Option 3 is \(\theta=241+360n\) (where \(n=k - 1\)), which is also correct. But maybe the question expects the formula starting with the original angle, so option 2, or the positive coterminal angle, option 3.

Wait, let's check the options again. The option \(\theta=-119 + 360^\circ(n)\) is option 2, and \(\theta=241 + 360^\circ(n)\) is option 3. Let's see the value of \(-119+360 = 241\), so both are equivalent. But let's check the problem's options. The correct options are \(\theta=-119 + 360^\circ(n)\) (option 2) and \(\theta=241 + 360^\circ(n)\) (option 3). But maybe the question considers the general formula as \(\theta=-119 + 360n\) (option 2) or \(\theta=241+360n\) (option 3).

Wait, let's calculate the coterminal angle:

If we have \(\theta=-119^\circ\), to find coterminal angles, we add or subtract \(360^\circ\) multiple times. So the general formula is \(\theta=-119^\circ+360^\circ n\), \(n\in\mathbb{Z}\) (option 2) and also \(\theta=241^\circ+360^\circ n\), \(n\in\mathbb{Z}\) (option 3) because \(-119+360 = 241\).

But let's check the options. The answer should be either option 2 or option 3. Wait, maybe the question has a mistake, but according to the formula, both are correct. But let's see the options again. The option \(\theta=-119 + 360n\) is directly from the formula, so that's the general formula based on the given angle.

Answer:

\(\boldsymbol{\theta=-119 + 360^\circ(n)}\) (corresponding to the option: \(\boldsymbol{\theta=-119 + 360^\circ(n)}\)) or \(\boldsymbol{\theta=241 + 360^\circ(n)}\) (corresponding to the option: \(\boldsymbol{\theta=241 + 360^\circ(n)}\)). But based on the formula application to the original angle, the answer is \(\theta=-119 + 360^\circ(n)\) (the second option).