Use the trigonometric substitution to write the algebraic...

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of $\theta,$ where $\mathbf{0}<\boldsymbol{\theta}<\pi / 2$
$$\sqrt{25-x^{2}}, \quad x=5 \sin \theta$$

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Key Concepts

Trigonometric Substitution

Trigonometric substitution is a method used in integration and algebraic manipulation to simplify expressions involving square roots. By substituting a trigonometric function for an algebraic expression, one can leverage trigonometric identities to simplify the radical. In the problem, substituting x = 5 sin ? transforms the algebraic expression under the square root into an expression involving cosine, which is easier to handle.

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is a fundamental trigonometric relation. After substituting x = 5 sin ?, the expression ?(25 ? x²) becomes ?(25(1 ? sin²?)), which simplifies to ?(25 cos²?). This simplification relies directly on the Pythagorean identity to replace 1 ? sin²? with cos²?.

Domain Considerations

When performing trigonometric substitution, it is important to consider the restrictions on the variable ? that ensure the validity of the substitution. For example, a domain such as 0 < ? < ?/2 guarantees that both sine and cosine functions are non-negative, which is crucial when simplifying square roots to their principal values.

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