Given a point on the unit circle at $ heta = frac{5pi}{6}$, find the coordinates of this point and determine the angle in degrees. Additionally, use the graphing calculator TI-Nspire to visualize the unit circle and verify the coordinates.
Answer 1
Ella Lewis
To solve the problem, follow these steps:
1. Identify the coordinates of the point on the unit circle at $\theta = \frac{5\pi}{6}$.
The coordinates can be determined using the unit circle definitions: $\left(\cos \theta, \sin \theta \right)$.
2. Calculate the coordinates:
$\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$
$\sin \frac{5\pi}{6} = \frac{1}{2}$
So, the coordinates are $\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$.
3. Convert the angle to degrees:
$\theta = \frac{5\pi}{6} \times \frac{180}{\pi} = 150^{\circ}$
4. Verify using TI-Nspire:
– Open the graphing calculator TI-Nspire.
– Plot the unit circle.
– Add a point at the angle $\theta = \frac{5\pi}{6}$ and verify the coordinates.
Final Answer: The coordinates are $\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$ and the angle is $150^{\circ}$.
Answer 2
Henry Green
Steps to solution:
1. Determine the coordinates on the unit circle at $ heta = frac{5pi}{6}$.
The coordinates are given by $left(cos heta, sin heta
ight)$.
2. Calculations:
$cos frac{5pi}{6} = -frac{sqrt{3}}{2}$
$sin frac{5pi}{6} = frac{1}{2}$
The coordinates are $left( -frac{sqrt{3}}{2}, frac{1}{2}
ight)$.
3. Conversion of angle:
$ heta = frac{5pi}{6} imes frac{180}{pi} = 150^{circ}$
4. Verification on TI-Nspire:
– Use the graphing tools to plot the unit circle.
– Add the point at $ heta = frac{5pi}{6}$ on the graph and check the coordinates.
Final Answer: Coordinates $left( -frac{sqrt{3}}{2}, frac{1}{2}
ight)$, Angle $150^{circ}$.
Answer 3
Mia Harris
1. Coordinates for $ heta = frac{5pi}{6}$ on the unit circle:
$left(cos heta, sin heta
ight) = left( -frac{sqrt{3}}{2}, frac{1}{2}
ight)$.
2. Angle in degrees:
$frac{5pi}{6} imes frac{180}{pi} = 150^{circ}$.
3. Verify on TI-Nspire by graphing the unit circle and plotting the point.
Answer: Coordinates $left( -frac{sqrt{3}}{2}, frac{1}{2}
ight)$, Angle $150^{circ}$.
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