-11.7.94 Find the rectangular equation

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In summary, the rectangular equation for the curve $r=\sin\left(\theta+\dfrac{\pi}{4}\right)$ is $\left(x-\dfrac{\sqrt{2}}{4}\right)^2 + \left(y - \dfrac{\sqrt{2}}{4}\right)^2 = \left(\dfrac{1}{2}\right)^2$, which represents a circle with radius $\dfrac{1}{2}$ centered at $\left(\dfrac{\sqrt{2}}{4}, \dfrac{\sqrt{2}}{4}\right)$. This can be derived by expanding the equation using trigonometric identities and completing the square.
  • #1
karush
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$\tiny{11.7.94 Kamahamai HS}$
Find the rectangular equation of the curve $r=\sin\left(\theta+\dfrac{\pi}{4}\right)$
$r=\sin \theta{\cos \dfrac{\pi}{4}
+{\cos \theta{\sin \dfrac{\pi}{4}}}}
=\sin \theta\left(\dfrac{\sqrt{2}}{2}\right)+\cos \theta\left(\dfrac{\sqrt{2}}{2}\right)
=\left(\dfrac{\sqrt{2}}{2}\right) (\sin \theta+\cos\theta)$

well so far anyway
Desmos plotted a circle
 
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  • #2
$r = \dfrac{\sqrt{2}}{2}(\cos{t}+\sin{t})$

$r^2 = \dfrac{\sqrt{2}}{2}(r\cos{t}+r\sin{t})$

$x^2+y^2 = \dfrac{\sqrt{2}}{2}(x+y)$

which leads to …

$\left(x-\dfrac{\sqrt{2}}{4}\right)^2 + \left(y - \dfrac{\sqrt{2}}{4}\right)^2 = \left(\dfrac{1}{2}\right)^2$
 
  • #3
so that's how you get a circle 🙄
https://dl.orangedox.com/QS7cBvdKw55RQUbliE
 
Last edited:

FAQ: -11.7.94 Find the rectangular equation

What does "-11.7.94" represent in the given rectangular equation?

The "-11.7.94" in the rectangular equation represents the x-coordinate of a point on the graph. It is a decimal number and is typically expressed as (x, y) in coordinate form.

How do you find the rectangular equation for a given point?

To find the rectangular equation for a point, you need to determine the x and y coordinates of the point. Then, you can plug these values into the equation x = -11.7.94 to get the rectangular equation in the form of (x, y).

Can the rectangular equation have negative coordinates?

Yes, the rectangular equation can have negative coordinates. In fact, the "-11.7.94" in the given equation indicates that the x-coordinate is negative. This means that the point is located to the left of the y-axis on the graph.

What is the significance of "-11.7.94" in the rectangular equation?

The "-11.7.94" in the rectangular equation represents the value of the x-coordinate for a specific point on the graph. It helps to locate the point on the x-axis and is an essential component in determining the equation of a line or curve.

How do you graph a point using the rectangular equation?

To graph a point using the rectangular equation, you need to plot the x and y coordinates on the graph. In this case, you would plot the point (-11.7.94, y) where y can be any value. This point would be located at -11.7.94 on the x-axis and at the corresponding y-value on the y-axis.

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