Introduction
The answer to this question is
\[\begin{align*} \boxed{\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}} \end{align*}\]
In this article, we’ll learn how to find \(\cos \dfrac{\pi}{4}\), which is the same as \(\cos 45^\circ\). We’ll approach this by using both a right triangle and the unit circle. After that, we’ll go through a practical example where knowing \(\cos \dfrac{\pi}{4}\) is helpful.
Converting \(\dfrac{\pi}{4}\) to degrees
To understand why \(\dfrac{\pi}{4}\) radians is equal to 45 degrees, let’s recall that a full circle is \(360^\circ\), which is also \(2\pi\) radians. And
\[\begin{align*} \pi \text{ radians} = 180^\circ\ \end{align*}\]
Dividing both sides by 4, we find:
\[\begin{align*} \dfrac{\pi}{4}\text{ radians} = 45^\circ \end{align*}\]
This means that \(\dfrac{\pi}{4}\) radians corresponds to a 45-degree angle.
Finding \(\cos \dfrac{\pi}{4}\) Using a Triangle
To understand \(\cos 45^\circ\), let’s look at a special right triangle called a 45-45-90 triangle, where both of the non-right angles are 45 degrees.
\[\begin{align*} \cos 45^\circ = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2} \end{align*}\]
Thus,
\[\begin{align*} \cos \dfrac{\pi}{4} = \cos 45^\circ = \dfrac{\sqrt{2}}{2} \end{align*}\]
Understanding the Unit Circle
The unit circle is a circle with a radius of 1 centered at the origin \((0, 0)\) on the coordinate plane. Each point on the unit circle represents the coordinates \((\cos \theta, \sin \theta)\) for an angle \(\theta\) measured counterclockwise from the positive \(x\)-axis. The unit circle helps us to find values of sine and cosine for any angle by looking at the coordinates of points on the circle. Review the Unit Circle article to understand it better.
Finding \(\cos \dfrac{\pi}{4}\) Using the Unit Circle
If you memorized the unit circle and remember that the \(x\)-coordinate of a point corresponds to the cosine of the angle, you can draw the diagram above and conclude that
\[\begin{align*} \boxed{\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}} \end{align*}\]
Example Problem Involving \(\cos \dfrac{\pi}{4}\)
Suppose a ladder is leaning against a wall, forming a $45^\circ$ angle with the ground. If the ladder is 10 feet long, how far is the base of the ladder from the wall?
Solution
When possible, you should always draw a diagram to help you solve the problem. In our case, the diagram looks like this:
Let the distance from the base of the ladder to the wall be \(x\). Since the angle between the ladder and the ground is 45 degrees, we can use the cosine function:
\[\begin{align*} \cos 45^\circ = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{x}{10} \end{align*}\]
Substitute \(\cos 45^\circ = \dfrac{\sqrt2}{2}\) and solve for \( x \):
\[\begin{align*} \dfrac{1}{2} = \dfrac{x}{10} \end{align*}\]
\[\begin{align*} x = 10 \cdot \dfrac{\sqrt2}{2} = \boxed{5\sqrt2 \text{ feet}} \end{align*}\]
