coordinates

coordinates. (Use variables as needed.) sin(θ) cos(θ) Need Help? Keea TtiwMrC SALGTRIG4 8.1.065. Convert the polar equation to rectangular coordinates. (Use variables as needed.) sin(θ) Need Help? Uizn WiaivMES Submit Answer SALGTRIG4 8.1.066. Convert the polar equation to rectangular coordinates. (Use variables as needed.) cos(θ)
SALGTRIG4 8.1.063. Convert the polar equation to rectangular coordinates. (Use variables as needed.) sin(θ) cos(θ) Need Help? Keea TtiwMrC SALGTRIG4 8.1.065. Convert the polar equation to rectangular coordinates. (Use variables as needed.) sin(θ) Need Help? Uizn WiaivMES Submit Answer SALGTRIG4 8.1.066. Convert the polar equation to rectangular coordinates. (Use variables as needed.) cos(θ)

The Correct Answer and Explanation is:

To convert polar equations into rectangular (Cartesian) coordinates, we use the following relationships between polar and rectangular coordinates:

1. r=x2+y2r = \sqrt{x^2 + y^2}r=x2+y2​
2. x=rcos⁡(θ)x = r \cos(\theta)x=rcos(θ)
3. y=rsin⁡(θ)y = r \sin(\theta)y=rsin(θ)

Now, let’s look at the given equations and how to convert them.

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1. sin⁡(θ)\sin(\theta)sin(θ) in Polar Coordinates

We are asked to convert sin⁡(θ)\sin(\theta)sin(θ) into rectangular coordinates.

1. Start with the relationship:
   y=rsin⁡(θ)y = r \sin(\theta)y=rsin(θ)
2. To express sin⁡(θ)\sin(\theta)sin(θ) in terms of xxx and yyy, divide both sides of the equation by rrr:
   sin⁡(θ)=yr\sin(\theta) = \frac{y}{r}sin(θ)=ry​
3. Since r=x2+y2r = \sqrt{x^2 + y^2}r=x2+y2​, substitute this into the equation:
   sin⁡(θ)=yx2+y2\sin(\theta) = \frac{y}{\sqrt{x^2 + y^2}}sin(θ)=x2+y2​y​

This is the conversion of sin⁡(θ)\sin(\theta)sin(θ) to rectangular coordinates.

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2. cos⁡(θ)\cos(\theta)cos(θ) in Polar Coordinates

Now, we are asked to convert cos⁡(θ)\cos(\theta)cos(θ) into rectangular coordinates.

1. Start with the relationship:
   x=rcos⁡(θ)x = r \cos(\theta)x=rcos(θ)
2. To express cos⁡(θ)\cos(\theta)cos(θ) in terms of xxx and yyy, divide both sides by rrr:
   cos⁡(θ)=xr\cos(\theta) = \frac{x}{r}cos(θ)=rx​
3. Again, substitute r=x2+y2r = \sqrt{x^2 + y^2}r=x2+y2​:
   cos⁡(θ)=xx2+y2\cos(\theta) = \frac{x}{\sqrt{x^2 + y^2}}cos(θ)=x2+y2​x​

This is the conversion of cos⁡(θ)\cos(\theta)cos(θ) to rectangular coordinates.

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3. sin⁡(θ)cos⁡(θ)\sin(\theta) \cos(\theta)sin(θ)cos(θ) in Polar Coordinates

Now, let’s convert sin⁡(θ)cos⁡(θ)\sin(\theta) \cos(\theta)sin(θ)cos(θ) into rectangular coordinates.

1. From the previous conversions:
   sin⁡(θ)=yx2+y2andcos⁡(θ)=xx2+y2\sin(\theta) = \frac{y}{\sqrt{x^2 + y^2}} \quad \text{and} \quad \cos(\theta) = \frac{x}{\sqrt{x^2 + y^2}}sin(θ)=x2+y2​y​andcos(θ)=x2+y2​x​
2. Multiply these two expressions together:
   sin⁡(θ)cos⁡(θ)=(yx2+y2)(xx2+y2)\sin(\theta) \cos(\theta) = \left( \frac{y}{\sqrt{x^2 + y^2}} \right) \left( \frac{x}{\sqrt{x^2 + y^2}} \right)sin(θ)cos(θ)=(x2+y2​y​)(x2+y2​x​)
3. Simplify the expression:
   sin⁡(θ)cos⁡(θ)=xyx2+y2\sin(\theta) \cos(\theta) = \frac{xy}{x^2 + y^2}sin(θ)cos(θ)=x2+y2xy​

This is the conversion of sin⁡(θ)cos⁡(θ)\sin(\theta) \cos(\theta)sin(θ)cos(θ) to rectangular coordinates.

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Conclusion

To summarize, here are the converted forms:

• sin⁡(θ)=yx2+y2\sin(\theta) = \frac{y}{\sqrt{x^2 + y^2}}sin(θ)=x2+y2​y​
• cos⁡(θ)=xx2+y2\cos(\theta) = \frac{x}{\sqrt{x^2 + y^2}}cos(θ)=x2+y2​x​
• sin⁡(θ)cos⁡(θ)=xyx2+y2\sin(\theta) \cos(\theta) = \frac{xy}{x^2 + y^2}sin(θ)cos(θ)=x2+y2xy​

These are the standard conversions used for polar to rectangular transformations. Let me know if you need further clarification!