Problem 70 - Precalculus: A Concise Course

Question

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7 .)\) $$ \sec (\arctan 3 x) $$

Step-by-Step Solution

Answer

The algebraic expression equivalent to \( \sec (\arctan 3x) \) is \( \sqrt{9x^2 + 1}\).

Step 1: Understanding the initial expression

Firstly, the expression is \( \sec (\arctan 3x) \). \(\sec(x)\) is the reciprocal of cosine function, and \(\arctan 3x\) gives us an angle for which the tangent is \( 3x \). Clear definition of every function will simplify the task.

Step 2: Applying arctangent expression

Next, in a right triangle, \(\tan (\theta)\) is equivalent to \( \frac {opposite}{adjacent} \). Therefore, considering the hint and variable substitution, let \( \arctan 3x = \theta \). Thus, \( 3x = \frac {opposite}{adjacent} \). If we consider the opposite side as \(3x\) and adjacent side as \(1\), using the Pythagorean theorem, the hypotenuse then will be \( \sqrt{(3x)^2 + 1^2} = \sqrt{9x^2 + 1} \).

Step 3: Applying the secant function

Lastly, as the secant function is the reciprocal of the cosine function, it is therefore the ratio of the length of the hypotenuse to the length of the adjacent side in a right triangle. Hence, plug in the length values into the secant function. For the expression \( \sec (\arctan 3x) \), it will be equivalent to \( \frac {hypotenuse}{adjacent} = \frac {\sqrt{9x^2 + 1}}{1} = \sqrt{9x^2 + 1}\).

Key Concepts

Right Triangle

A right triangle is a basic concept in geometry where one of the internal angles is exactly 90 degrees, which makes the other two angles complementary. Right triangles have certain properties that make them useful in trigonometry. The sides are classified as follows:

• Hypotenuse: The longest side, opposite the right angle.
• Opposite Side: The side opposite the angle you are working with.
• Adjacent Side: The side next to the angle you are concerned with, excluding the hypotenuse.

These sides form the basis of defining trigonometric functions like sine, cosine, and tangent. The Pythagorean theorem is a fundamental principle when working with right triangles, stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides: \( a^2 + b^2 = c^2 \). This theorem allows us to calculate the length of any side if the other two are known, which is crucial in solving for trigonometric values.

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions and is derived from the cosine function. Specifically, it is the reciprocal of the cosine function:\[\sec \theta = \frac{1}{\cos \theta}\].In a right triangle, it can be described as the ratio between the length of the hypotenuse and the length of the adjacent side. This means:\[\sec \theta = \frac{hypotenuse}{adjacent}\].It is particularly useful in converting angles to a more tangible understanding based on right triangle geometry. Understanding secant in context with cosine helps in visualizing and solving problems that require reciprocals of trigonometric functions, essential in higher-level calculations like trigonometric identities and solving equations. For the given problem, once the right triangle is constructed, the secant function helps determine the final expression.

Arctangent Function

The arctangent function, often written as \( \arctan(x) \), is the inverse of the tangent function. It helps in determining the angle whose tangent is a given number:\[\theta = \arctan (opposite/adjacent)\].In a right triangle, if you know the ratio of the lengths of the opposite side to the adjacent side, you can find the angle using the arctangent function. For example, in this problem, \( \arctan(3x) \) describes an angle where the tangent is \( 3x \), meaning the opposite side is \( 3x \) and the adjacent side is \( 1 \).Using this information, you can construct a triangle and then use related functions to find other trigonometric values involving this angle. Understanding the application of arctangent is critical in transitioning from known side ratios to angles, enabling comprehensive solutions in trigonometric contexts.