Problem 30 - Thomas Calculus

Question

Evaluate the expressions in Exercises \(29-40\) $$ \sec \left(\tan ^{-1} 2 x\right) $$

Step-by-Step Solution

Answer

\( \sqrt{4x^2 + 1} \)

Step 1: Understanding the Inverse Trigonometric Function

The expression involves the inverse tangent function, \( \tan^{-1}(2x) \). Let \( \theta = \tan^{-1}(2x) \), which means \( \tan(\theta) = 2x \). This is the angle \( \theta \) whose tangent is \( 2x \).

Step 2: Setting Up a Right Triangle

Visualize a right triangle where \( \theta \) is one of the non-right angles. According to \( \tan(\theta) = 2x \), the opposite side is \( 2x \) and the adjacent side is \( 1 \). This is because tangent is the ratio of the opposite to the adjacent side.

Step 3: Using the Pythagorean Theorem

Calculate the hypotenuse \( h \) using the Pythagorean theorem: \[ h = \sqrt{(2x)^2 + 1^2} = \sqrt{4x^2 + 1} \]

Step 4: Determining the Secant

Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}} \). From the triangle, \( \sec(\theta) = \frac{\sqrt{4x^2 + 1}}{1} = \sqrt{4x^2 + 1} \).

Step 5: Evaluating the Expression

The expression \( \sec(\tan^{-1}(2x)) \) evaluates to \( \sqrt{4x^2 + 1} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles based on given trigonometric values. Consider the function \( \tan^{-1}(2x) \), also known as the arctangent function. This function gives us an angle whose tangent is \( 2x \). We use the notation \( \theta = \tan^{-1}(2x) \), meaning the tangent of the angle \( \theta \) is equal to \( 2x \).

• Arctangent Function: Returns an angle \( \theta \) such that \( \tan(\theta) = 2x \).
• Range: The principal value of \( \tan^{-1}(x) \) is in the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\).
• Application: Allows us to work backwards from the tangent value to determine angle size, helpful in geometry and calculus.

Inverse functions are essential for solving problems where angles are not given initially, allowing the discovery of unknown angles from known trigonometric values.

Right Triangle

A right triangle is a geometric shape featuring one 90-degree angle. When working with trigonometric functions, right triangles provide a visual method to determine ratios between the sides.
In our problem, the right triangle is set up based on the angle \( \theta \) for \( \tan(\theta) = 2x \). The parts of the triangle are:

• Opposite Side: Value is \( 2x \), representing the side opposite angle \( \theta \).
• Adjacent Side: Value is 1, the side next to angle \( \theta \), excluding the hypotenuse.
• Hypotenuse: To be determined using the Pythagorean theorem, which connects all side lengths of a triangle.

Right triangles are essential in trigonometry as they define the primary trigonometric ratios: sine, cosine, and tangent. By assigning side lengths to opposite, adjacent, and hypotenuse sides, each ratio represents a unique relationship between these sides.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially when dealing with right triangles. It states that in a right triangle, the square of the hypotenuse length is the sum of the squares of the other two sides.
For our right triangle where \( \tan(\theta) = 2x \):

• Formula: \( h = \sqrt{(2x)^2 + 1^2} \)
• Calculation: Evaluate \( h \) which yields \( \sqrt{4x^2 + 1} \)
• Use: The hypotenuse is essential for calculating other trigonometric functions, like secant and cosine.

The theorem helps in deriving the length of the hypotenuse, ensuring all side ratios are accurately used for further trigonometric calculations, like obtaining \( \sec(\theta) \). This theorem makes sure measurements derived from a triangle's angles are reliable, grounding trigonometric and geometric principles effectively.