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 x2y2=ay=ay=xx2+(ya)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
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+(ya)2=a2 x^2 + (y - a)^2 = a^2
5. (xa)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

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