Problem 99 - Precalculus with Limits

Question

In Exercises 93 - 104, use the trigonometric substitution tow rite the algebraic expression as a trigonometric function of \( \theta \), where \( 0 < \theta < \pi/2 \). \( \sqrt{x^2 + 25} \), \( x = 5 \tan \theta \)

Step-by-Step Solution

Answer

The algebraic expression \( \sqrt{x^2 + 25} \) can be expressed as a trigonometric function \( 5 \sec \theta \) where \( x = 5 \tan \theta \) and \( 0 < \theta < \pi/2 \).

Step 1: Substitution

Starting with \( x = 5 \tan \theta \), substitute \( x \) in the given expression. The new expression becomes \( \sqrt{(5 \tan \theta)^2 + 25} \).

Step 2: Simplification

Simplify the squared term inside the square root to get \( \sqrt{25 \tan^2 \theta + 25} \).

Step 3: Factor Out

Factor out a 25 from the terms inside the square root to get \( \sqrt{25( \tan^2 \theta + 1)} \). Using the Pythagorean identity \( \tan^2 \theta + 1 = \sec^2 \theta \), you obtain \( \sqrt{25 \sec^2 \theta} \).

Step 4: Square Root

Square root of 25 and \( \sec^2 \theta \) separately to get \( 5 \lvert \sec \theta \rvert \). However, we are given \( 0 < \theta < \frac{\pi}{2} \). So, \( \sec \theta \) is positive in this interval. Therefore, you can drop the absolute value to obtain the final expression \( 5 \sec \theta \).

Key Concepts

Pythagorean Identities

Pythagorean identities are fundamental relationships involving trigonometric functions, reminding us of the Pythagorean Theorem from geometry. These identities express the idea that for a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. In trigonometry, these identities help relate different functions.

One of the core Pythagorean identities is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity shows how sine and cosine are interconnected and is useful in simplifying equations.

Another important identity is:

• \( \tan^2 \theta + 1 = \sec^2 \theta \)

This was used in the solution to transform \( \tan^2 \theta + 1 \) into \( \sec^2 \theta \), making it easier to simplify the expression. Knowing and understanding these identities allow us to manipulate and simplify trigonometric expressions fluently.

Trigonometric Functions

Trigonometric functions describe the relationships between the angles and sides of triangles. They are essential for solving problems involving periodic phenomena.

Here are some of the standard trigonometric functions:

• Sine \((\sin \theta)\): Opposite side over hypotenuse in a right triangle.
• Cosine \((\cos \theta)\): Adjacent side over hypotenuse.
• Tangent \((\tan \theta)\): Opposite side over adjacent side, or \( \frac{\sin \theta}{\cos \theta} \).
• Cosecant, Secant, Cotangent: These are the reciprocals of sine, cosine, and tangent, respectively.

Each function concerns a ratio of specific sides within a right triangle and varies as the angle changes. In solving the exercise, we dealt with \(\tan\theta\) and \(\sec\theta\), where \(x\) was set as \(5 \tan\theta\). By substituting \(\tan\theta\) and using simplification techniques, we ultimately expressed everything in terms of \(\sec\theta\). Understanding these functions is instrumental in trigonometry as they allow for expressing complex geometric phenomena in mathematical terms.

Trigonometric Interval

The trigonometric interval is a range within which a given trigonometric function operates. In our exercise, the interval \(0 < \theta < \frac{\pi}{2}\) plays a crucial role in determining the nature of the function values.

This interval corresponds to the first quadrant of the unit circle, where all trigonometric functions are positive. Here's why it's important:

• Sine and Cosine: Both functions are positive in this interval, reflecting the acute angles associated with right triangles.
• Tangent and Secant: Since tangent is the ratio of sine to cosine, both having positive values, tangent is also positive.
  Secant, being the reciprocal of cosine, follows suit in positiveness.

This property allowed us to simplify \( 5 \lvert \sec \theta \rvert \) to just \( 5 \sec \theta \), since \( \sec \theta \) is positive in this interval. Understanding these intervals helps in determining the sign and value of functions, thus ensuring exact and simplified results. Knowing the basics of these intervals is incredibly helpful for more advanced trigonometric studies.