Use the relationships x = r cos Ø and y = r sin Ø to write the equation x² - y² = 1in terms of r and Ø.

Question

please.

Answer 1

Keep this identity in mind:
(cos(x))^2 - (sin(x))^2 = cos(2x)

x² - y² = 1

(rcosØ)^2 - (rsinØ)^2 = 1
r^2 (cosØ)^2 - r^2 (sinØ)^2 = 1
r^2 [(cosØ)^2 - (sinØ)^2] = 1
r^2 (cos2Ø) = 1

Answer 2

plugging in we get

(r cos Ø)^2 - (r sin Ø)^2 = 1 in terms of r and Ø.
r^2 [(cos Ø)^2 - (sin Ø)^2] =1

the answer above satisfies the requirements but you can simplify further by using the sin^2+cos^2=1 identity to get:
r^2 [(cos Ø)^2 -(1- cos Ø)^2] =1
r^2 [2(cos Ø)^2 -1] =1
is the simplest form

Answer 3

x² - y² = 1
(r cos Ø)² - (r sin Ø)² = 1
r² cos² Ø - r² sin² Ø = 1
r² (cos² Ø - sin² Ø) = 1
r² (cos 2Ø) = 1

Answer 4

x ^2 - y ^2 = 1

( r cos☺)^2 - (r sin☺)^2 = 1

r^2 cos^2 ☺ - r^2 sin^2 ☺ = 1

r^2 (cos^2 ☺ - sin^2 ☺) = 1

r^2 ( cos^2 ☺) = 1

(cos^2 ☺) = 1/r^2

(cos ☺)^2 = 1/r^2 (square root both
sides)

cos ☺ = 1/r

☺ = cos^ -1 (1/r)

Answer 5

(using x instead of theta as i cant get that symbol up and also using ^2 for squared)

(r^2cos^2x) - (r^2sin^2x) = 1

r^2 ( cos ^2 x - sin ^2 x) = 1

r^2 (cos x + sin x)(cos x - sin x) = 1

Answer 6

x^2-y^2=1
r^2cos^2 a-r^2sinâ=1, a=phi
r^2cos 2a=1ANS.

Answer 7

r^2cos^2@-r^2sin^2@=1

Answer 8

x2 - y2 = 1
r2 cos2 (phi) - r2 sin2(phi) = 1
r2 cos(2phi) = 1

There you go...