Problem 98 - Precalculus : Building Concepts and Connections

Question

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\cos s=\cos t\) $$t=\frac{\pi}{2}$$

Step-by-Step Solution

Answer

The angle \(s\) that satifies \(\cos s = \cos t\) and \(s \neq t\) when \(0 \leq s<2 \pi\) and \(t=\frac{\pi}{2}\) is \(s = \frac{3\pi}{2}\).

Step 1: Identify starting point

First, we have \(t=\frac{\pi}{2}\) and we need to find \(s\) such that \(\cos s = \cos t\). The cosine value of \(t\) would therefore be: \[\cos t = \cos \frac{\pi}{2}\]

Step 2: Compute cosine value

We calculate \(\cos \frac{\pi}{2}\), which is 0.

Step 3: Find values of s

From the unit circle, we know that \(\cos s = 0\) when \(s = \frac{\pi}{2}\) or \(s = \frac{3\pi}{2}\). However, since \(s \neq t\) we exclude \(s = \frac{\pi}{2}\), therefore \[s = \frac{3\pi}{2}\]

Key Concepts

The Cosine Function

The cosine function is a fundamental trigonometric function that helps us to describe the relationship between the angles and sides of a right-angled triangle. But its importance extends beyond triangles to modeling waves, oscillations, and even in defining the coordinates of points on a circle.

Using the unit circle, which is a circle with a radius of 1 unit, we can represent angles and their corresponding cosine values. The cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle. So, when dealing with problems involving the cosine function, we often refer to the unit circle to find our values.

For instance, if we have an angle like \(t = \frac{\pi}{2}\), the cosine function relates to this angle by telling us how far along the x-axis the point is. Since the point for \(t = \frac{\pi}{2}\) is at the top of the circle, its distance along the x-axis is 0, which is indeed the value of \(\cos t\). Understanding this concept allows students to tackle a wide variety of trigonometry problems involving the cosine function.

The Unit Circle

The unit circle is a foundational concept not just in trigonometry but in all of mathematics. It’s a simple idea—a circle with a radius of one unit centered at the origin of a coordinate plane—but it packs a lot of power for solving problems.

On the unit circle, any point can be defined using the coordinates \((\cos \theta, \sin \theta)\), where \(\theta\) is an angle measured from the positive x-axis. The beauty of the unit circle is that it allows us to visualize trigonometric functions and easily find corresponding values for sine, cosine, and tangent functions for angles in all four quadrants.

The provided exercise demonstrates this by using the unit circle to find an angle \(s\) that has the same cosine value as another angle \(t\). Since the cosine function gives us the x-coordinate, we look for places where the circle intersects with the same x-value as that of \(\cos t\). This visualization on the unit circle makes comprehending complex trigonometric relationships much more intuitive.

Radians

Radians are the standard unit of angular measure used in many areas of mathematics. Unlike degrees, which divide a circle into 360 arbitrary units, radians are based on the radius of the circle. A radian is the angle that, when centered at the circle's origin, subtends an arc equal to the radius of the circle.

Understanding radians is crucial when working with trigonometric functions and the unit circle. There are \(2\pi\) radians in a full circle, so an angle of \(\frac{\pi}{2}\) radians, like the one mentioned in the exercise, corresponds to a quarter of a full rotation around the circle. This angle leads us directly up the y-axis, which means the x-coordinate (or cosine value) is zero.

Radians provide a more natural way to measure angles in the context of the unit circle, and they often simplify calculations in trigonometry. Getting comfortable with converting between degrees and radians and visualizing angles in radians helps students to more profoundly understand and solve trigonometry problems.