Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

Therefore, the algebraic expression \(\sqrt{25-x^2}\) can be written as a trigonometric function of \(\theta\) where \(0 \leq \theta < \pi/2\) as \(5 \cos \theta\).

Step 1: Trigonometric Substitution

Start by replacing \(x\) in \(\sqrt{25-x^2}\) with \(5\sin\theta\). It becomes \(\sqrt{25-(5\sin\theta)^2}\).

Step 2: Simplification

Simplify the equation above, \(\sqrt{25-(5\sin\theta)^2}\) becomes \(\sqrt{25-25\sin^2\theta} = \sqrt{25(1-\sin^2\theta)}\). Using the Pythagorean identity \(1-\sin^2\theta = \cos^2\theta\), the expression becomes \(\sqrt{25\cos^2\theta}\).

Step 3: Further Simplification

Further simplify the equation, \(\sqrt{25\cos^2\theta}\) will turn into \(5\cos\theta\). Here, it is assumed that \(\cos\, \theta\) is always positive as \(\theta\) is in the first quadrant.

Key Concepts

Pythagorean Identity

In trigonometry, the Pythagorean Identity is a crucial concept. It stems directly from the Pythagorean Theorem in geometry. This identity states that for any angle \( \theta \), the square of the sine function plus the square of the cosine function always equals one: \[\sin^2\theta + \cos^2\theta = 1.\]This identity is fundamental when dealing with trigonometric substitutions, as it allows us to express one trigonometric function in terms of another.
This was precisely used in the solution to simplify \( \sqrt{25(1 - \sin^2\theta)} \) into \( \sqrt{25\cos^2\theta} \) by recognizing that \( 1- \sin^2\theta = \cos^2\theta \). This step is powerful, as it directly transforms the complexity of square roots with trigonometric terms into a more familiar and manageable form. Always remember this identity as a key tool in your mathematical toolkit.

Trigonometric Functions

Trigonometric functions, namely sine, cosine, and tangent, are fundamental in mathematics. They derive from the relationships between the angles and sides of right-angled triangles.
In our context, we focus on the sine and cosine functions.

• The sine of an angle \( \theta \), denoted as \( \sin\theta \), is the ratio of the length of the side opposite the angle to the hypotenuse.
• Similarly, the cosine of \( \theta \), denoted as \( \cos\theta \), is the ratio of the adjacent side to the hypotenuse.

Included in our exercise is the trigonometric substitution, \( x = 5 \sin\theta \), which strategically uses these functions to simplify expressions involving square roots.
Understanding these basic trigonometric functions prepares you for transforming complex mathematical expressions into comprehensible ones.

First Quadrant

Angles in trigonometry are conventionally divided into four quadrants. The first quadrant is especially important because here, both sine and cosine functions have positive values.
When an angle \( \theta \) lies in the first quadrant, typically defined as \( 0 < \theta < \pi/2 \), it means that

• \( \sin\theta \) is positive.
• \( \cos\theta \) is positive.

This fact plays a vital role in simplifications, like in our solution.
When we simplified \( \sqrt{25\cos^2\theta} \) to \( 5\cos\theta \), we relied on knowing that \( \cos\theta \) remains positive in this range. This ensures no additional negative factors affect the resulting expression.
Understanding the properties of angles in the first quadrant helps maintain the integrity of trigonometric expressions during simplification and transformation processes.