6. Using the trigonometric substitution x = 8 sec θ, where x ≥ 8 and 0 < θ ≤ π/2, express tan θ in terms of x.

Verified step by step guidance

1. Step 1: Recall the trigonometric identity for secant: $sec(\theta )\; =\; x/8$. This comes from the substitution $x\; =\; 8\; sec(\theta )$.
2. Step 2: Use the Pythagorean identity for trigonometric functions: $tan²(\theta )\; =\; sec²(\theta )\; -\; 1$. Substitute $sec(\theta )\; =\; x/8$ into this identity.
3. Step 3: Simplify the expression for $tan²(\theta )$: $tan²(\theta )\; =\; (x/8)²\; -\; 1$. Expand and simplify further.
4. Step 4: Take the square root of both sides to find $tan(\theta )$: $tan(\theta )\; =\; \surd ((x/8)²\; -\; 1)$. Ensure the square root is valid for the given domain of $\theta$.
5. Step 5: Conclude that $tan(\theta )$ is expressed in terms of $x$ as $tan(\theta )\; =\; \surd ((x²/64)\; -\; 1)$.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots. By substituting a variable with a trigonometric function, we can transform complex expressions into more manageable forms. In this case, substituting x with 8 sec θ allows us to express the integral in terms of θ, which can simplify the integration process.

Secant Function

The secant function, denoted as sec θ, is the reciprocal of the cosine function, defined as sec θ = 1/cos θ. It is particularly useful in trigonometric identities and substitutions. In the context of the substitution x = 8 sec θ, it helps relate the variable x to the angle θ, facilitating the conversion of trigonometric expressions into algebraic forms.

Tangent Function

The tangent function, tan θ, is defined as the ratio of the opposite side to the adjacent side in a right triangle, or equivalently, tan θ = sin θ/cos θ. In the context of the substitution x = 8 sec θ, we can express tan θ in terms of x by using the identity tan θ = sin θ / (1/cos θ) = sin θ * sec θ. This relationship is crucial for solving problems involving angles and their corresponding trigonometric values.