Question
Given that cosθ=−3√2
, find the coordinates of the point where the terminal side of θ
intersects the unit circle, if θ
is an obtuse angle.(1 point)
Responses
(−3√2,−12)
left parenthesis negative Start Fraction Start Root 3 End Root over 2 End Fraction comma negative Start Fraction 1 over 2 End Fraction right parenthesis
(−3√2,12)
left parenthesis negative Start Fraction Start Root 3 End Root over 2 End Fraction comma Start Fraction 1 over 2 End Fraction right parenthesis
(12,−3√2)
left parenthesis Start Fraction 1 over 2 End Fraction comma negative Start Fraction Start Root 3 End Root over 2 End Fraction right parenthesis
(−3√2,12√)
, find the coordinates of the point where the terminal side of θ
intersects the unit circle, if θ
is an obtuse angle.(1 point)
Responses
(−3√2,−12)
left parenthesis negative Start Fraction Start Root 3 End Root over 2 End Fraction comma negative Start Fraction 1 over 2 End Fraction right parenthesis
(−3√2,12)
left parenthesis negative Start Fraction Start Root 3 End Root over 2 End Fraction comma Start Fraction 1 over 2 End Fraction right parenthesis
(12,−3√2)
left parenthesis Start Fraction 1 over 2 End Fraction comma negative Start Fraction Start Root 3 End Root over 2 End Fraction right parenthesis
(−3√2,12√)
Answers
GPT-4o mini
To find the coordinates where the terminal side of the angle θ intersects the unit circle, we start with the given information:
1. We have \(\cos \theta = -\frac{3}{\sqrt{2}}\) (noting that there may be a notation issue—this is not a standard value for cosine in the unit circle).
2. Since θ is an obtuse angle, we know that \(\theta\) is between 90° and 180°.
The coordinates of any point on the unit circle can be expressed in terms of the cosine and sine of the angle:
\[
(x, y) = (\cos \theta, \sin \theta)
\]
However, because we need to find the sine value as well, we will have to use the identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Given \(\cos \theta = -\frac{3}{\sqrt{2}}\), we substitute this into the identity:
\[
\sin^2 \theta + \left(-\frac{3}{\sqrt{2}}\right)^2 = 1
\]
Calculating the square of the cosine:
\[
\sin^2 \theta + \frac{9}{2} = 1
\]
Now, solve for \(\sin^2 \theta\):
\[
\sin^2 \theta = 1 - \frac{9}{2} = -\frac{7}{2}
\]
This is a negative value, which indicates that we may have had a misinterpretation with \(\cos \theta = -\frac{3}{\sqrt{2}}\) because on the unit circle, cosines must lie within [-1, 1].
Let’s check if instead you meant \(\cos \theta = -\frac{\sqrt{2}}{2}\):
In this case, \(\cos \theta = -\frac{\sqrt{2}}{2}\).
Using the identity:
\[
\sin^2 \theta + \left(-\frac{\sqrt{2}}{2}\right)^2 = 1
\]
Calculating the cosine square:
\[
\sin^2 \theta + \frac{1}{2} = 1
\]
Now, solve for \(\sin^2 \theta\):
\[
\sin^2 \theta = 1 - \frac{1}{2} = \frac{1}{2}
\]
Then, taking the square root gives us:
\[
\sin \theta = \pm \frac{\sqrt{2}}{2}
\]
Since θ is obtuse, we choose the positive value:
\[
\sin \theta = \frac{\sqrt{2}}{2}
\]
Thus, the coordinates where the terminal side of angle θ intersects the unit circle are:
\[
\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)
\]
Now, comparing it with the provided options and realizing there's an error in the choices, the correct coordinates based on typical values would be approximately \((-1, 0)\) and similar values near the unit circle if revised.
However, among the given choices to select:
- It does not look to correspond directly.
Thus, please revise the initial cos θ value if there is a misunderstanding.
1. We have \(\cos \theta = -\frac{3}{\sqrt{2}}\) (noting that there may be a notation issue—this is not a standard value for cosine in the unit circle).
2. Since θ is an obtuse angle, we know that \(\theta\) is between 90° and 180°.
The coordinates of any point on the unit circle can be expressed in terms of the cosine and sine of the angle:
\[
(x, y) = (\cos \theta, \sin \theta)
\]
However, because we need to find the sine value as well, we will have to use the identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Given \(\cos \theta = -\frac{3}{\sqrt{2}}\), we substitute this into the identity:
\[
\sin^2 \theta + \left(-\frac{3}{\sqrt{2}}\right)^2 = 1
\]
Calculating the square of the cosine:
\[
\sin^2 \theta + \frac{9}{2} = 1
\]
Now, solve for \(\sin^2 \theta\):
\[
\sin^2 \theta = 1 - \frac{9}{2} = -\frac{7}{2}
\]
This is a negative value, which indicates that we may have had a misinterpretation with \(\cos \theta = -\frac{3}{\sqrt{2}}\) because on the unit circle, cosines must lie within [-1, 1].
Let’s check if instead you meant \(\cos \theta = -\frac{\sqrt{2}}{2}\):
In this case, \(\cos \theta = -\frac{\sqrt{2}}{2}\).
Using the identity:
\[
\sin^2 \theta + \left(-\frac{\sqrt{2}}{2}\right)^2 = 1
\]
Calculating the cosine square:
\[
\sin^2 \theta + \frac{1}{2} = 1
\]
Now, solve for \(\sin^2 \theta\):
\[
\sin^2 \theta = 1 - \frac{1}{2} = \frac{1}{2}
\]
Then, taking the square root gives us:
\[
\sin \theta = \pm \frac{\sqrt{2}}{2}
\]
Since θ is obtuse, we choose the positive value:
\[
\sin \theta = \frac{\sqrt{2}}{2}
\]
Thus, the coordinates where the terminal side of angle θ intersects the unit circle are:
\[
\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)
\]
Now, comparing it with the provided options and realizing there's an error in the choices, the correct coordinates based on typical values would be approximately \((-1, 0)\) and similar values near the unit circle if revised.
However, among the given choices to select:
- It does not look to correspond directly.
Thus, please revise the initial cos θ value if there is a misunderstanding.