Question

Explain which trig functions and identities produce circles, ellipses and hyperbolas. Will give Brainliest for correct answer.

Answer 1

Answer:

Circle: sin²(theta) + cos²(theta) = 1

Ellipse: sin²(theta) + cos²(theta) = 1

Hyperbola: sinh²(theta) - cosh²(theta) = 1

Step-by-step explanation:

Circle

Cartesian form from polar form:

x = rcos(theta)

y = rsin(theta)

x² + y² = r²sin²(theta) + r²cos²(theta)

x² + y² = r²(sin²(theta) + cos²(theta))

Using sin² + cos² = 1, we get

x² + y² = r²

Ellipse

x = acos(theta)

y = bsin(theta)

cos(theta) = x/a

sin(theta) = y/b

x²/a² + y²/b² = sin²(theta) + cos²(theta)

x²/a² + y²/b² = 1

Hyperbola

x = rcosh(theta)

y = rsinh(theta)

x²/r² - y²/r² = sinh²(theta) - cosh²(theta)

x²/r² - y²/r² = 1

Answer 2

Answer:

The general equation for any conic section is:

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,

where A, B, C, D, E, F are fixed coefficients and A, B, C are not simultaneously equal to 0.

But, we can also write equations for conic section in terms of trigonometric functions using Parametric Equations:

For circle:

(x, y) = (rcos(theta), rsin(theta)) <=> x^2 + y^2 = r^2,

because  (cos(theta))^2 + (sin(theta))^2 = 1

For ellipse:

(x, y) = (acos(theta), bsin(theta)) or x^2/a^2 + y^2/b^2 = 1

because  (cos(theta))^2 + (sin(theta))^2 = 1

For hyperbola:

(x, y) = (asec(theta), btan(theta)) or  x^2/a^2 - y^2/b^2= 1

because  (cos(theta))^2 + (sin(theta))^2 = 1

Hope this helps!

:)