Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec x=2$$

Step-by-Step Solution

Answer

The solutions of the equation \( \sec x = 2 \) in the interval \([0, 2\pi )\) are \(x=\frac{\pi}{3}\) and \(x=\frac{5\pi}{3}\).

Step 1: Convert secant to cosine

Replace \(\sec x\) by \(\frac{1}{\cos x}\) in the equation, to get \( \frac{1}{\cos x} = 2 \). This equation can be further rewritten as: \(\cos x = \frac{1}{2}\).

Step 2: Identify the angles

The values of \(x\) for which \(\cos x= \frac{1}{2}\) in the interval \([0, 2\pi )\) are \(x=\frac{\pi}{3}\) and \(x=\frac{5\pi}{3}\).

Step 3: Verify the solutions

Plug these angles back into the original equation and make sure the left-hand side equals 2.

Key Concepts

Secant Function

The secant function, denoted as \( \text{sec}(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function, which means \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \). Understanding the secant function begins with a solid grasp of the cosine function.
In a right triangle, the cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse. When we take the reciprocal of this ratio, we get the secant function. The secant function is undefined when the cosine function is zero because you cannot divide by zero. This happens at angles of \( \frac{\text{π}}{2} \) and \( \frac{3\text{π}}{2} \) for a standard unit circle.
When solving equations involving the secant function, it's often useful to convert secants into cosines, a more familiar function, which makes it easier to solve the equations. As in the exercise, \( \text{sec}(x) = 2 \) becomes \( \text{cos}(x) = \frac{1}{2} \) when we take the reciprocal of both sides. This transformation utilizes the reciprocal identity of secant and cosine and is a key strategy in finding solutions to trigonometric equations that involve secant.

Cosine Function

The cosine function is an essential component in the toolbox of trigonometry. It relates the angle of a right triangle to the ratio of the length of the triangle's adjacent side to its hypotenuse. The cosine function is written as \( \text{cos}(x) \).
This function is especially significant as it appears in the definition of other trigonometric functions, including the secant function mentioned earlier. As seen in the textbook exercise, we might encounter an equation like \( \text{cos}(x) = \frac{1}{2} \). To solve such an equation, one would find the specific angle(s) that produce the value of \( \frac{1}{2} \) when the cosine function is applied. These angles, also known as inverse cosine values, are found using the unit circle or trigonometric tables. In the interval \( [0, 2\text{π}) \), the cosine function has a value of \( \frac{1}{2} \) at \( x = \frac{\text{π}}{3} \) and \( x = \frac{5\text{π}}{3} \), showcasing symmetry in a circle. Learning to identify these key angles is a pivotal skill in trigonometry.

Trigonometric Identities

In trigonometry, identities are equations that hold true for all values for which both sides are defined. Trigonometric identities are important because they allow us to simplify expressions and solve equations that involve trigonometric functions.
One of the basic sets of identities is the reciprocal identities, which associate functions like sine, cosine, and tangent with their reciprocals cosecant, secant, and cotangent, respectively. For example, the one used in our textbook exercise is \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \).
Another important set is the Pythagorean identities, which relate the square of sine and cosine functions to one. There are also angle sum and difference identities that express the trigonometric functions of an angle composed of two separate angles. Mastering these identities is fundamental to becoming proficient in trigonometry as they provide a framework for manipulating and solving a wide range of problems.

Trigonometry Interval Solutions

When solving trigonometric equations, it's essential to pay attention to the given interval since trigonometric functions have periodic properties and thus can have infinitely many solutions.
In our exercise, the solutions must be within the interval \( [0, 2\text{π}) \), which represents one complete cycle around the unit circle. It's critical to identify on the unit circle where the trigonometric function takes the given value within the specified interval. This knowledge helps in finding what are called 'principal angle' solutions, or the smallest positive angles that satisfy the equation.
For equations involving the cosine or secant function, it is possible to have two solutions within one cycle that have the same value but occur at different points along the circle, reflecting the symmetry of the cosine function. Good practice involves verifying that the solutions you've found indeed fall within the given interval and checking them back into the original equation to confirm their validity. Understanding the concept of interval solutions is crucial for accurate problem-solving in trigonometry.