Questions: For the given values of an angle α, find the terminal point P(x, y) on the unit circle. (a) If α = π/2, then x = and y = ; (b) If α = -π/2, then x = and y = (c) If α = π/3, then x = and y = ; (d) If α = -π/3, then x = and y = (e) If α = 3π/4, then x = and y = (f) If α = -3π/4, then x = and y =

For the given values of an angle α, find the terminal point P(x, y) on the unit circle.
(a) If α = π/2, then x = and y = ;
(b) If α = -π/2, then x = and y =
(c) If α = π/3, then x = and y = ;
(d) If α = -π/3, then x = and y =
(e) If α = 3π/4, then x = and y =
(f) If α = -3π/4, then x = and y =

Transcript text: For the given values of an angle $\alpha$, find the terminal point $P(x, y)$ on the unit circle. (a) If $\alpha=\frac{\pi}{2}$, then $x=$ $\square$ and $y=$ $\square$ ; (b) If $\alpha=-\frac{\pi}{2}$, then $x=$ $\square$ and $y=$ $\square$ (c) If $\alpha=\frac{\pi}{3}$, then $x=$ $\square$ and $y=$ $\square$ ; (d) If $\alpha=-\frac{\pi}{3}$, then $x=$ $\square$ and $y=$ $\square$ (e) If $\alpha=\frac{3 \pi}{4}$, then $x=$ $\square$ and $y=$ $\square$ (f) If $\alpha=-\frac{3 \pi}{4}$, then $x=$ $\square$ and $y=$ $\square$

Solution

Solution Steps

To find the terminal point $ P(x, y) $ on the unit circle for a given angle $\alpha$, we use the fact that the coordinates of the terminal point are given by $ (\cos(\alpha), \sin(\alpha)) $.

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate system. For any angle $\alpha$, the coordinates of the terminal point $P(x, y)$ on the unit circle are given by $ (\cos(\alpha), \sin(\alpha)) $.

Step 2: Calculating Coordinates for Given Angles

We need to find the coordinates for the following angles:

$\alpha = \frac{\pi}{2}$
$\alpha = -\frac{\pi}{2}$
$\alpha = \frac{\pi}{3}$

Step 3: Using Trigonometric Functions

For each angle, we use the cosine and sine functions to determine the coordinates:

For $\alpha = \frac{\pi}{2}$: \[ x = \cos\left(\frac{\pi}{2}\right) = 0.0000 \] \[ y = \sin\left(\frac{\pi}{2}\right) = 1.0000 \]
For $\alpha = -\frac{\pi}{2}$: \[ x = \cos\left(-\frac{\pi}{2}\right) = 0.0000 \] \[ y = \sin\left(-\frac{\pi}{2}\right) = -1.0000 \]
For $\alpha = \frac{\pi}{3}$: \[ x = \cos\left(\frac{\pi}{3}\right) = 0.5000 \] \[ y = \sin\left(\frac{\pi}{3}\right) = 0.8660 \]

Final Answer

\[ \boxed{ \begin{array}{ll} (a) & x = 0, \; y = 1 \\ (b) & x = 0, \; y = -1 \\ (c) & x = \frac{1}{2}, \; y = \frac{\sqrt{3}}{2} \\ (d) & x = \frac{1}{2}, \; y = -\frac{\sqrt{3}}{2} \\ (e) & x = -\frac{\sqrt{2}}{2}, \; y = \frac{\sqrt{2}}{2} \\ (f) & x = -\frac{\sqrt{2}}{2}, \; y = -\frac{\sqrt{2}}{2} \\ \end{array} } \]

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