91Ó°ÊÓ

The Pythagorean Theorem is a fundamental principle in geometry. It relates to right triangles, stating that, in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be formulated as:\[a^2 + b^2 = c^2\]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other sides.
In the trigonometric substitution exercise, this theorem helps us find the third side of the triangle when given that one side is represented as \(x = 8 \cos \theta\) and the hypotenuse is \(8\). From this, we derive:\[\text{remaining side} = \sqrt{8^2 - x^2} = \sqrt{64 - x^2}\]This step sets the foundation for further trigonometric simplification.

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable involved. These identities are essential for simplifying trigonometric expressions and solving equations. In our example, the Pythagorean identity for sine and cosine plays a vital role:\[\sin^2 \theta + \cos^2 \theta = 1\]By recognizing that \(x = 8 \cos \theta\), we substitute into the expression for the third side to get \[\sqrt{1 - \cos^2 \theta} = \sqrt{\sin^2 \theta}\]This identity allows us to replace the expression under the square root explicitly with \(\sin^2 \theta\), leading to a simpler form to work with. Without understanding these identities, simplification would be challenging.

The cosine function, often represented as \(\cos\), is a fundamental trigonometric function. It originates from a right triangle, where the cosine of an angle \(\theta\) is the ratio of the length of the adjacent side to the hypotenuse:\[\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\]In our exercise, \(x = 8 \cos \theta\) indicates that \(x\) acts as the adjacent side to the angle \(\theta\), with the hypotenuse having a length of 8. Recognizing this relationship helps to correctly apply trigonometric substitution, starting from an algebraic format to a trigonometric identity, enabling further simplification steps. The cosine function's role here is pivotal in transforming the given expression into a form suitable for actual evaluation.

The sine function, expressed as \(\sin\), is another core trigonometric function. It relates the opposite side of a right triangle to its hypotenuse:\[\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\]Once we simplify the expression under the square root using the Pythagorean identity, we arrive at \[\sqrt{1 - \cos^2 \theta} = \sqrt{\sin^2 \theta} = \sin \theta\]Considering \(\theta\) is in the first quadrant, where all trigonometric functions are positive, we can safely assert: \[8\sin\theta = 8 |\sin \theta|\]By understanding the sine function and its properties, we effectively transform and reach at the simplest trigonometric expression to represent the original algebraic expression. This final step ties together the use of trigonometric substitution to convert challenging radicals into more approachable forms.