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Convert r=sinθ, 1.6 Minutes and Done!
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To convert r=sinθ, multiply both sides by r, giving you r² = r*sinθ. You should know that r² = x² + y², and r*sinθ = y. Rearrange the equation by gathering terms x² + y² - y = 0.
Complete the square on the y-term. Add and subtract (1/2)² = 1/4 to maintain balance. The equation becomes x² + (y - 1/2)² - 1/4 = 0.
Simplify further (x - 0)² + (y - 1/2)² = 1/4.
Now, let's find the center and radius of the circle. The center is (0, 1/2), and the radius is 1/2.
If you need a visual representation, draw a circle with the center at (0, 1/2) and a radius of 1/2. It should look balanced. Leave a like if you found this helpful!
Exercise 1
Convert the equation r = cosθ to rectangular form.
Solution
To convert r = cosθ, follow the same steps as before. Multiply both sides by r, giving you r² = r*cosθ. Knowing that r² = x² + y² and r*cosθ = x, rearrange the equation as follows x² + y² - x = 0.
Exercise 2
Convert the equation r = 2sinθ to rectangular form.
Solution
Multiply both sides by r r² = 2r*sinθ. Since r² = x² + y² and r*sinθ = y, the equation becomes x² + y² - 2y = 0.
Exercise 3
Convert the equation r = 3sinθ + cosθ to rectangular form.
Solution
Multiply both sides by r r² = 3r*sinθ + r*cosθ. Substitute r² = x² + y², r*sinθ = y, and r*cosθ = x into the equation x² + y² = 3y + x.
Complete the square on the y-term. Add and subtract (1/2)² = 1/4 to maintain balance. The equation becomes x² + (y - 1/2)² - 1/4 = 0.
Simplify further (x - 0)² + (y - 1/2)² = 1/4.
Now, let's find the center and radius of the circle. The center is (0, 1/2), and the radius is 1/2.
If you need a visual representation, draw a circle with the center at (0, 1/2) and a radius of 1/2. It should look balanced. Leave a like if you found this helpful!
Exercise 1
Convert the equation r = cosθ to rectangular form.
Solution
To convert r = cosθ, follow the same steps as before. Multiply both sides by r, giving you r² = r*cosθ. Knowing that r² = x² + y² and r*cosθ = x, rearrange the equation as follows x² + y² - x = 0.
Exercise 2
Convert the equation r = 2sinθ to rectangular form.
Solution
Multiply both sides by r r² = 2r*sinθ. Since r² = x² + y² and r*sinθ = y, the equation becomes x² + y² - 2y = 0.
Exercise 3
Convert the equation r = 3sinθ + cosθ to rectangular form.
Solution
Multiply both sides by r r² = 3r*sinθ + r*cosθ. Substitute r² = x² + y², r*sinθ = y, and r*cosθ = x into the equation x² + y² = 3y + x.