Studdy AI Logo
Math  /  Geometry

QuestionConvert the following polar coordinates to rectangular coordinates equations. (Here aa is a constant number.) r=ar=a □\square r=asec⁡(θ)r=a \sec (\theta) □\square r=acsc⁡(θ)r=a \csc (\theta) □\square r=2asin⁡(θ)r=2 a \sin (\theta) □\square r=2acos⁡(θ)r=2 a \cos (\theta) □\square r=asec⁡(2θ)r=a \sec (2 \theta) □\square θ=π4\theta=\frac{\pi}{4} □\square
Drag or tap the options below to fill in the blanks x2−y2=ay=ay=xx2+(y−a)2=a2x=a)2+y2=a2x2+y2=a2\left.x^{2}-y^{2}=a y=a y=x x^{2}+(y-a)^{2}=a^{2} x=a\right)^{2}+y^{2}=a^{2} x^{2}+y^{2}=a^{2}

Studdy Solution

STEP 1

1. The polar coordinates are given in the form r=f(θ) r = f(\theta) .
2. The conversion from polar to rectangular coordinates involves using the relationships x=rcos⁡(θ) x = r \cos(\theta) and y=rsin⁡(θ) y = r \sin(\theta) .

STEP 2

1. Convert r=a r = a to rectangular coordinates.
2. Convert r=asec⁡(θ) r = a \sec(\theta) to rectangular coordinates.
3. Convert r=acsc⁡(θ) r = a \csc(\theta) to rectangular coordinates.
4. Convert r=2asin⁡(θ) r = 2a \sin(\theta) to rectangular coordinates.
5. Convert r=2acos⁡(θ) r = 2a \cos(\theta) to rectangular coordinates.
6. Convert r=asec⁡(2θ) r = a \sec(2\theta) to rectangular coordinates.
7. Convert θ=π4 \theta = \frac{\pi}{4} to rectangular coordinates.

STEP 3

Convert r=a r = a to rectangular coordinates:
Using the identity x2+y2=r2 x^2 + y^2 = r^2 , we substitute r=a r = a :
x2+y2=a2 x^2 + y^2 = a^2

STEP 4

Convert r=asec⁡(θ) r = a \sec(\theta) to rectangular coordinates:
Since sec⁡(θ)=1cos⁡(θ) \sec(\theta) = \frac{1}{\cos(\theta)} , we have rcos⁡(θ)=a r \cos(\theta) = a .
Thus, x=a x = a .

STEP 5

Convert r=acsc⁡(θ) r = a \csc(\theta) to rectangular coordinates:
Since csc⁡(θ)=1sin⁡(θ) \csc(\theta) = \frac{1}{\sin(\theta)} , we have rsin⁡(θ)=a r \sin(\theta) = a .
Thus, y=a y = a .

STEP 6

Convert r=2asin⁡(θ) r = 2a \sin(\theta) to rectangular coordinates:
Multiply both sides by r r to get r2=2arsin⁡(θ) r^2 = 2a r \sin(\theta) .
Substitute r2=x2+y2 r^2 = x^2 + y^2 and rsin⁡(θ)=y r \sin(\theta) = y :
x2+y2=2ay x^2 + y^2 = 2a y

STEP 7

Convert r=2acos⁡(θ) r = 2a \cos(\theta) to rectangular coordinates:
Multiply both sides by r r to get r2=2arcos⁡(θ) r^2 = 2a r \cos(\theta) .
Substitute r2=x2+y2 r^2 = x^2 + y^2 and rcos⁡(θ)=x r \cos(\theta) = x :
x2+y2=2ax x^2 + y^2 = 2a x

STEP 8

Convert r=asec⁡(2θ) r = a \sec(2\theta) to rectangular coordinates:
Using the identity sec⁡(2θ)=1cos⁡(2θ) \sec(2\theta) = \frac{1}{\cos(2\theta)} , we have rcos⁡(2θ)=a r \cos(2\theta) = a .
This is more complex to convert directly, but typically involves using double angle identities and further algebraic manipulation.

STEP 9

Convert θ=π4 \theta = \frac{\pi}{4} to rectangular coordinates:
Using the identity tan⁡(θ)=yx \tan(\theta) = \frac{y}{x} , for θ=π4 \theta = \frac{\pi}{4} , we have tan⁡(π4)=1 \tan\left(\frac{\pi}{4}\right) = 1 .
Thus, y=x y = x .
The rectangular coordinates equations are:
1. x2+y2=a2 x^2 + y^2 = a^2
2. x=a x = a
3. y=a y = a
4. x2+(y−a)2=a2 x^2 + (y - a)^2 = a^2
5. (x−a)2+y2=a2 (x - a)^2 + y^2 = a^2
6. Further manipulation needed for r=asec⁡(2θ) r = a \sec(2\theta)
7. y=x y = x

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord