Problem 43 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\) $$ \sec \theta=1 $$

Step-by-Step Solution

Answer

\(\theta = 0\)

Step 1: Convert into Cosine

Given that \(\sec \theta = 1\), convert this into a cosine function since \(\sec \theta = 1/\cos \theta\). So the equation becomes \(\cos \theta = 1\).

Step 2: Find \(\theta\) values

To solve for \(\theta\), we need to ask where \(\cos \theta = 1\) between \(0 \leq \theta < 2 \pi\). Our solution is \(\theta=0\) and \(\theta=2 \pi\). However, since \(2 \pi\) is not within our given range, we exclude it. So our only solution is \(\theta = 0\).

Key Concepts

Secant Function

In trigonometry, the secant function is often seen as a bit mysterious, largely because it's less common than its trigonometric cousins like sine, cosine, and tangent. But its role is essential and straightforward to understand. The secant function, denoted as \(\sec \theta\), is defined as the reciprocal of the cosine function. This means: \[ \sec \theta = \frac{1}{\cos \theta} \] So, whenever you encounter a secant problem, the first step is usually to express it in terms of cosine. This helps in simplifying the equation. For example, when you have \(\sec \theta = 1\), turning this into \( \cos \theta = 1 \) will make it much easier to solve.Remember, since secant is the reciprocal of cosine, it can never be zero because division by zero is undefined. Thus, if you're working with secant, your focus is on the non-zero values of cosine.

Cosine Function

The cosine function is key in trigonometry as it provides the 'adjacent over hypotenuse' ratio in a right triangle. It is one of the most frequently used trigonometric functions, denoted by \(\cos \theta\). Its value ranges from -1 to 1, making it ideal for various applications in mathematics and physics.To solve equations with cosine, you often look for angles where it takes specific values. For our problem, \(\cos \theta = 1\), this situation arises at particular angles on the unit circle. When working with cosine, you should be familiar with its basic values at standard angles, like 0, \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\).For \(\cos \theta = 1\), the angle \(\theta\) is 0 radians (or equivalent positions, like \(2\pi\), though they might be out of range in specific problems). Learning these values helps you solve trigonometric equations more comfortably.

Unit Circle

The unit circle is a crucial concept in understanding trigonometric functions such as secant and cosine. It's a circle with a radius of one, centered at the origin in a coordinate plane. This simple yet powerful tool allows us to define trigonometric functions for any angle, not just those within a right triangle.On the unit circle, each angle \(\theta\) corresponds to a point \((x, y)\), where \(x = \cos \theta\) and \(y = \sin \theta\). This means that the x-coordinate gives the cosine value of \(\theta\) directly. For the equation \(\cos \theta = 1\), look for points on the unit circle where the x-coordinate is 1. The unit circle greatly simplifies solving trigonometric equations because it visually represents the cyclical nature of trigonometric functions. It also highlights symmetries and periodic behaviors, which are fundamental in trigonometry. Using the unit circle helps you quickly identify valid angles that solve equations like \(\cos \theta = 1\) across different intervals like \(0 \leq \theta < 2\pi\).