Question

Write the expression in algebraic form. \(\sec [\arcsin (x-1)]\)

Short answer

The expression in algebraic form, \(\sec [\arcsin (x-1)]\) is either \(1/\sqrt{1 - (x - 1)^2}\) or \(-1/\sqrt{1 - (x - 1)^2}\) depending on the quadrant of the angle.

Step-by-step solution

Define the inverse sine

The inverse sine function, \(\arcsin(x)\), yields an angle whose sine is \(x\). Thus, \(\arcsin(x-1)\) denotes an angle whose sine is \(x-1\). Let's denote this angle by \(y\), so \(y = \arcsin(x-1)\) and \(\sin(y) = x-1\).

Apply Pythagoras’ theorem

By using the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\), we have \(\cos^2(y)=1- \sin^2(y)=1 - (x - 1)^2\). Hence, \(\cos(y)\) is equal to either \(\sqrt{1 - (x - 1)^2}\) or \(-\sqrt{1 - (x - 1)^2}\), depending on the quadrant in which \(y\) lies.

Find the secant

The secant function, \(\sec(\theta)\), is the reciprocal of the cosine function, \(\cos(\theta)\). Hence \(\sec(y) = 1/\cos(y)\).

Substitue the values

Substituting the expression for \(\cos(y)\) from Step 2 into the definition of \(\sec(y)\) gives \(\sec [\arcsin (x-1)] = 1/\sqrt{1 - (x - 1)^2}\) or \(\sec [\arcsin (x-1)] = -1/\sqrt{1 - (x - 1)^2}\) depending upon the quadrant of the angle.

Key concepts

Arcsin

Understanding the arcsin function is a starting point when dealing with inverse trigonometric problems. When you see an expression like \( \arcsin(x) \), it refers to the angle whose sine value equals \( x \). Think of \( \arcsin \) as asking the question: 'What angle gives me this sine value?'.

When working with \( \arcsin(x-1) \), it implies we are looking for an angle, let's call it \( y \), such that the sine of \( y \) is equal to \( x-1 \). Mathematical notation for this relationship would be \( \sin(y) = x-1 \). It is important to note that \( \arcsin \) will always give you an angle that is between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) (or -90 and +90 degrees), which corresponds to the range of the sine function for which it is uniquely invertible.

Secant Function

Going deeper into our problem, the next step involves understanding the secant function, often abbreviated as \( \sec \). In contrast to the more commonly known cosine, \( \sec(\theta) \) is the reciprocal of the cosine of \( \theta \). Therefore, whenever you see \( \sec(\theta) \), you can read it as \( \frac{1}{\cos(\theta)} \).

The secant function becomes very useful when dealing with triangles in which working with the hypotenuse and the adjacent side directly is more convenient. However, it's essential to always remember that the secant function is undefined whenever the cosine function is zero, which happens at odd multiples of \( \frac{\pi}{2} \).

Pythagorean Identity

Connection with the Circle

The Pythagorean identity is pivotal in trigonometry. At its core, it states that for any angle \( \theta \), the square of the sine plus the square of the cosine of \( \theta \) always equals one: \( \sin^2(\theta) + \cos^2(\theta) = 1 \). This relationship stems directly from the Pythagorean theorem and is a reflection of the unit circle's properties, where the radius (hypotenuse) is always one.

When considering an angle \( y = \arcsin(x-1) \), we can use the Pythagorean identity to find \( \cos(y) \) as the problem requires. Recognising that \( \sin(y) = x-1 \) and applying the identity, we end up with \( \cos^2(y) = 1 - (x-1)^2 \).

Why Two Values for Cosine?

We have two potential values for the cosine because of the nature of positive and negative square roots. The actual value depends on \( y \)'s quadrant, which we can usually infer from the context of the problem.

Algebraic Expression

Finally, let's address the term algebraic expression. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \( x \) or \( y \)), and operators (such as addition and subtraction). Converting trigonometric expressions into algebraic form often involves using identities, properties, and known values to simplify or rewrite the expression in terms of basic algebra.

For our original exercise, \( \sec [\arcsin (x-1)] \), the algebraic expression we seek is based on the secant function and the angle given by the inverse sine. By substituting the values from the Pythagorean identity and considering the reciprocal nature of the secant function, we reach an algebraic expression that represents the trigonometric statement without any trigonometric functions. This process is essential in calculus and higher mathematics, where the algebraic form can be manipulated more easily.