a sec theta + b tan theta =c and p sec theta + q sec theta = r then eliminate theta
To eliminate theta from the given equations, we need to manipulate the equations in a way that allows us to cancel out the trigonometric functions involving theta.
First, let's look at the first equation:
a sec(theta) + b tan(theta) = c
By using the trigonometric identity sec^2(theta) = 1 + tan^2(theta), we can rewrite tan(theta) as (sec^2(theta) - 1):
a sec(theta) + b (sec^2(theta) - 1) = c
Next, let's simplify the equation by collecting terms:
b sec^2(theta) + (a - b) sec(theta) - c = 0
Now, let's move on to the second equation:
p sec(theta) + q sec(theta) = r
We can factor out sec(theta) from both terms:
(sec(theta))(p + q) = r
Finally, we have two equations involving sec(theta) that we can solve simultaneously:
b sec^2(theta) + (a - b) sec(theta) - c = 0 (Equation 1)
(sec(theta))(p + q) = r (Equation 2)
To eliminate theta, we can substitute sec(theta) from Equation 2 into Equation 1. Let's solve Equation 2 for sec(theta):
sec(theta) = r / (p + q)
Now, substitute sec(theta) in Equation 1 with the value we just found:
b (r / (p + q))^2 + (a - b)(r / (p + q)) - c = 0
This equation can be further simplified and solved for the value of r in terms of a, b, c, p, and q.
a secθ + b tanθ = c
p secθ + q secθ = r
I suspect a typo. Is that what you meant? That means that secθ = r/(p+q)