Write the trigonometric expression as an algebraic expression in u.

sine left parenthesis secant Superscript negative 1 Baseline u right parenthesissinsec−1u




To write the trigonometric expression \(\sin(\sec^{-1}(u))\) as an algebraic expression in \(u\), follow these steps:

### Step 1: Definition of \(\sec^{-1}(u)\)
The expression \(\sec^{-1}(u)\) represents the angle \(\theta\) such that:
\[
\sec(\theta) = u \quad \text{or equivalently} \quad \frac{1}{\cos(\theta)} = u
\]
Thus, we have:
\[
\cos(\theta) = \frac{1}{u}
\]

### Step 2: Use of Pythagorean Identity
We know that:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Substituting \(\cos(\theta) = \frac{1}{u}\) into this equation:
\[
\sin^2(\theta) + \left(\frac{1}{u}\right)^2 = 1
\]
\[
\sin^2(\theta) + \frac{1}{u^2} = 1
\]
\[
\sin^2(\theta) = 1 - \frac{1}{u^2}
\]
\[
\sin(\theta) = \sqrt{1 - \frac{1}{u^2}}
\]

### Step 3: Final Expression
Since \(\theta = \sec^{-1}(u)\), we have:
\[
\sin(\sec^{-1}(u)) = \sqrt{1 - \frac{1}{u^2}}
\]

Thus, the algebraic expression for \(\sin(\sec^{-1}(u))\) is:
\[
\boxed{\sqrt{1 - \frac{1}{u^2}}}
\]

Would you like further details or have any questions on this?

### 5 Related Questions:
1. What is the geometric interpretation of \(\sec^{-1}(u)\)?
2. How do you express \(\cos(\sec^{-1}(u))\) as an algebraic expression in \(u\)?
3. How would you differentiate \(\sin(\sec^{-1}(u))\) with respect to \(u\)?
4. What is the domain of the function \(\sin(\sec^{-1}(u))\)?
5. How does this process change if we are dealing with other inverse trigonometric functions?

**Tip**: Inverse trigonometric functions often yield angles, so rewriting them using right-triangle identities can help simplify expressions.

Write the trigonometric expression as an algebraic expression in u: sin(sec⁻¹(u))

Learn how to rewrite the trigonometric expression sin(sec⁻¹(u)) as an algebraic expression in u using inverse trigonometric functions and the Pythagorean identity. Follow the step-by-step solution suitable for Grades 10-12.

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