Problem 35 - Calculus: One and Several Variables

Question

Find the number(s) \(x\) in the interval \([0.2 \pi]\) which satisfy the equation. $$\cos x=\sqrt{2} / 2$$.

Step-by-Step Solution

Answer

The values of \(x\) that satisfy the equation \(\cos x = \sqrt{2} / 2\) in the interval \([0, 2 \pi]\) are \(x = \pi / 4\) and \(x = 7 \pi / 4\).

Step 1: Understand the Cosine Function

Cosine function is the horizontal coordinate of a point on the unit circle. On the unit circle, each radian corresponds to an angle measured from the positive x-axis. Here, we are interested in when the horizontal coordinate (cosine) equals \(\sqrt{2} / 2\).

Step 2: Identify the Radians with Cosine Equals \(\sqrt{2} / 2\)

The cosine of \(45°\) or \(\pi / 4\) radians is \(\sqrt{2} / 2\). Hence, \(\cos (\pi / 4) = \sqrt{2} / 2\). However, cosine function is also equal to \(\sqrt{2} / 2\) at \(7\pi / 4\) radians because cosine function is positive in the fourth quadrant too. Hence, \(\cos (7\pi / 4) = \sqrt{2} / 2\). These angles are where \(\cos x = \sqrt{2} / 2\) on the unit circle.

Step 3: Find the Values of \(x\) in the Given Interval

Finally, we need to identify the solutions in the interval \([0, 2\pi]\). Both \(\pi / 4\) and \(7 \pi / 4\) fall within this interval, so they are the solutions for the given equation.

Key Concepts

Understanding the Unit Circle

Imagine a circle with a radius of 1 unit centered at the origin of a coordinate system. This is known as the unit circle, an essential concept in trigonometry. Each point on the unit circle corresponds to an angle measured from the positive x-axis, and the coordinates of that point are defined by the cosine and sine of that angle.

For example, if you were to start at the point (1,0) on the unit circle and trace a path counter-clockwise, the angle you travel through corresponds to the arc's length because the radius is 1. When we speak of the cosine function in relation to the unit circle, we're referring to the x-coordinate of these points. In the context of the exercise, the value given for the cosine function, \(\sqrt{2} / 2\), represents the horizontal distance from the origin to where our angle's line intersects the unit circle.

Radians as a Measure of Angles

Radians provide a way of measuring angles based on the arc length divided by the radius of a circle. This is a more natural measure in mathematics than degrees because it relates directly to the distance traveled around a circle. For every angle, there is an equivalent measure in radians, where \(2\pi\) radians is equal to 360 degrees, meaning \(\pi\) radians equals 180 degrees.

It's essential to understand radians to be able to work with trigonometric functions effectively. For the exercise, when it mentions an interval of \( [0, 2\pi] \), it's referring to all angle measures that fall within a complete circular rotation. Radians make it easier to identify specific angles on the unit circle, such as \(\pi / 4\) and \(7\pi / 4\), which correspond to the angles whose cosine value is \(\sqrt{2} / 2\).

Cosine of an Angle

The cosine of an angle in trigonometry represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle, or in the context of the unit circle, it is the x-coordinate of a point where the terminal side of the angle intersects the circle.

In our exercise, finding the angle where \(\cos x = \sqrt{2} / 2\) involves identifying the point on the unit circle's circumference that has an x-coordinate of \(\sqrt{2} / 2\). As cosine is positive in both the first and fourth quadrants, the angles of \(\pi / 4\) and \(7\pi / 4\) radians are the solutions. It's crucial to realize that cosine repeats its values periodically due to the circular nature of its graph, hence the function's multiple solutions for certain values.