Question

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0<θ<π/2. xx2−9​​,x=3sec(θ)

Answer 1

To make the trigonometric substitution x = 3sec(θ) in the expression √(x^2 - 9), we substitute x and simplify the result using trigonometric identities, ultimately obtaining the simplified expression 3tan(θ).

Step-by-step explanation:

The student is asking how to make a trigonometric substitution into an algebraic expression and then simplify it. Specifically, the substitution given is x = 3sec(θ) for the expression √(x^2 - 9) in the context of a trigonometry problem. To substitute and simplify, we first replace x with 3sec(θ) in the expression:

√(x^2 - 9) becomes √((3sec(θ))^2 - 9). Simplifying inside the square root gives us √(9sec^2(θ) - 9). Factoring out 9 from the square root leads to √(9(sec^2(θ) - 1)). Since sec^2(θ) - 1 can be rewritten as tan^2(θ) using the Pythagorean identity for secant (sec^2(θ) = 1 + tan^2(θ)), the expression simplifies to √(9tan^2(θ)). Taking the square root of this expression gives us 3tan(θ), which is the simplified result of the trigonometric substitution.