Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.
x = sec Q
y = cos Q
x^2 + y^2 = 1/cos^2 + sin^2/cos^2 =
x^2(1 +sin^2) = x^2(2-cos^2)
x^2(2-1/x^2) = 2x^2 - 1
x^2 - y^2 = 1
My teacher said to use secant*cosine = 1. Please explain!!
Isn't sec x = 1/cosx ??
so isn't (secQ)(cosQ)
= (1/cosQ)(cosQ) = 1 ???
so we know
secQ(cosQ) = 1
xy=1
looks like we got our equation!
To eliminate the parameter Q and write the corresponding rectangular equation, you need to express x and y in terms of each other. In this case, we have x = sec(Q) and y = cos(Q).
First, recall that the secant function is the reciprocal of the cosine function: sec(Q) = 1/cos(Q).
Next, we can substitute this expression for x into the equation x^2 - y^2 = 1:
(1/cos(Q))^2 - cos(Q)^2 = 1
Simplifying this equation, we get:
1/cos^2(Q) - cos^2(Q) = 1
Using the identity cos^2(Q) + sin^2(Q) = 1, we can rewrite cos(Q)^2 as 1 - sin(Q)^2:
1/cos^2(Q) - (1 - sin(Q)^2) = 1
Now, simplify the equation further:
1/cos^2(Q) - 1 + sin^2(Q) = 1
Multiply through by cos^2(Q) to get rid of the fraction:
1 - cos^2(Q) + sin^2(Q)*cos^2(Q) = cos^2(Q)
Rearrange the terms:
cos^2(Q) - sin^2(Q)*cos^2(Q) = 1
Factor out cos^2(Q):
cos^2(Q)(1 - sin^2(Q)) = 1
Using the Pythagorean identity sin^2(Q) + cos^2(Q) = 1, we can simplify this equation further:
cos^2(Q)*cos^2(Q) = 1
cos^4(Q) = 1
Now, taking the square root of both sides, we have:
cos^2(Q) = ±1
Finally, by recalling that cos^2(Q) = x^2 and sin^2(Q) = 1 - cos^2(Q), we can rewrite the equation as:
x^2 - y^2 = 1
Therefore, the rectangular equation whose graph represents the curve is x^2 - y^2 = 1.
Using the identity secant times cosine equals 1 (sec(Q) * cos(Q) = 1), we were able to eliminate the parameter Q and obtain the rectangular equation.