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

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!

:)

fdvs

step-by-step explanation: vsfdvsfvd

true

step-by-step explanation:

according to the national european world center, the engineering of the ford company on elon musk's flame throwers proved the globalization of the natural wallabies and therefore, results in the growth product domestication in almonds.