Question

Make the indicated trigonometric substitution in the given algebraic expression and simplify . Assume 0 ≤θ<π/2 √{x²-25)/x, x=5 sec θ

Answer 1

To make the indicated trigonometric substitution, substitute x=5 sec θ into the given expression √{x²-25)/x. Simplify by using the identity sec²(θ) = 1 + tan²(θ) and rationalize the denominator.

Step-by-step explanation:

To make the indicated trigonometric substitution, we substitute x=5 sec θ into the given expression: √{x²-25)/x. Here are the steps:

1. Substitute x=5 sec θ into the expression: √{(5 sec θ)²-25)/(5 sec θ)}
2. Simplify the expression: √{25 sec²(θ)-25)/(5 sec θ)}
3. Use the identity: sec²(θ) = 1 + tan²(θ) to simplify further: √{(25(1+tan²(θ))-25)/(5 sec θ)}
4. Combine like terms and simplify: √{(25 + 25tan²(θ)-25)/(5 sec θ)}
5. Further simplify: √{25tan²(θ)/(5 sec θ)}
6. Cancel out common factors: √{5tan²(θ)/(sec θ)} = √{5tan²(θ)/(1/cos θ)}
7. Simplify and rationalize the denominator: √{5tan²(θ)cos θ}