Question

Find \(\left(f^{-1}\right)(a)\) for the function \(f\) and real number \(a\). $$ f(x)=\cos 2 x, \quad 0 \leq x \leq \frac{\pi}{2} \quad \quad a=1 $$

Short answer

So, \(f^{-1}(1) = 0\).

Step-by-step solution

Identify the Function and The Required Points

The given function is \(f(x)=\cos(2x)\) and we are given the point \(a=1\).

Find the Inverse of the Function

The inverse of the cosine function is arccosine function. To find the inverse of \(f(x)\), we first replace \(f(x)\) by \(y\), i.e. \(y=\cos(2x)\). Then we switch \(x\) and \(y\), yielding \(x=\cos(2y)\). To isolate \(y\), we take the arccosine of each side, which gives us \(y = \frac{\arccos(x)}{2}\). Therefore, \(f^{-1}(a) = \frac{\arccos(a)}{2}\).

Substitute the Point Into the Inverse Function

Now we substitute \(a=1\) into \(f^{-1}(a)\) to get: \(f^{-1}(1) = \frac{\arccos(1)}{2}\). Arccosine of 1 equals to 0, which result in: \(f^{-1}(1) = \frac{0}{2} = 0\).

Key concepts

Arccosine Function

The arccosine function, often written as \( \arccos \), plays a fundamental role in trigonometry when we're looking to revert the process of cosine function. The term comes from 'arc cosine,' reflecting the function's connection to the arc length on a unit circle corresponding to a given cosine value.

The arccosine of a number is the angle whose cosine is that number. This function assumes values between 0 and \( \pi \) radians, or between 0 and 180 degrees. For example, \( \arccos(1) \) is 0 because the cosine of 0 is 1. Understanding the arccosine function helps you to work backward from a cosine value to the corresponding angle, which is crucial for solving many problems in trigonometry.

It's also worth noting that the arccosine function is a type of inverse trigonometric function and is key in solving trigonometric equations where you need to find the angle for a given trigonometric value. Furthermore, remember that the domain for the arccosine function is [-1, 1], since the cosine of an angle can never exceed this interval.

Finding Inverse Functions

The process of finding inverse functions involves switching the roles of the input and output of the original function. This mathematical procedure is akin to tracing your steps back in a maze to find the start point after reaching the end.

For a function \( f(x) \), the inverse function, denoted by \( f^{-1}(x) \), will give us the value of \( x \) that resulted in a particular output. When finding an inverse function, we typically swap the dependent and independent variables and then solve for the new dependent variable. However, a function must be one-to-one - each x value has a unique y value - to have an inverse that is also a function.

In the example from the exercise, \( y = \cos(2x) \) reverses to \( x = \cos(2y) \) and then to \( y = \frac{\arccos(x)}{2} \), thus creating our inverse function. It's important to apply this concept with care since functions can sometimes be restricted to ensure they have an inverse that is also a function. Understanding how to competently find inverse functions is a crucial skill for solving many algebraic and geometric problems.

Trigonometry

Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles. Fundamentally, it is built upon the trigonometric functions like sine, cosine, and tangent, which relate the angles of a right triangle to the ratios of its sides.

These trigonometric functions are vital in various areas of study and practical applications including geometry, engineering, physics, and even areas outside of STEM like music and art. For instance, in the realm of architecture, trigonometry allows for the precise calculation of angles and distances for constructing structurally sound and visually appealing buildings.

The key to mastering trigonometry is understanding the properties and applications of trigonometric functions. This involves learning how to graph these functions, solve equations involving them, and leverage their inverse counterparts. The ability to navigate these concepts allows you to decode complex problems and apply trigonometry effectively.