Question

Write the expression in algebraic form. \(\sec (\arctan 4 x)\)

Short answer

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

Step-by-step solution

Use the definition of arctan

Arctangent is defined as the angle whose tangent is the given number. If \( \arctan 4x = y \), then \( \tan y = 4x \).

Draw a right triangle

Draw a right triangle such that the angle at the origin is y. By the definition of the tangent function, which is 'opposite/adjacent', the side opposite to angle y can be taken as \(4x\) (which is the given tangent of y) and the adjacent side as 1. The hypotenuse, using Pythagoras theorem is \(\sqrt{1+(4x)^2}\)

Find the secant

The secant of an angle y in a right triangle is defined as the 'hypotenuse/adjacent side'. From the triangle in step 2, \(\sec y = \frac{\sqrt{1+(4x)^2}}{1}\)

Substituting y

Since \( \arctan 4x = y \), we need to substitute y in the expression found in step 3. Hence, \(\sec (\arctan 4 x) = \sqrt{1+(4x)^2}\)