Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the multiple-angle equation. $$\sec 4 x=2$$

Step-by-Step Solution

Answer

\[ x = \frac{\pi}{12} + \frac{n\pi}{2} , \frac{5\pi}{12} + \frac{n\pi}{2} \] where n is an integer.

Step 1: Convert secant to cosine

Sec(x) is reciprocal of Cos(x). So, change the given secant equation to cosine form, \( \cos 4x = 1/2 \).

Step 2: Solve for cos 4x

The cos function equals to 1/2 at \( \pi/3 \) and \( 5\pi/3 \) in 0 to 2\(\pi\). So, \( 4x = \pi/3 + 2n\pi \) and \( 4x = 5\pi/3 + 2n\pi \), where n is any integer.

Step 3: Solve for x

Rearranging the equations results in \( x = \pi/12 + n\pi/2 \) and \( x = 5\pi/12 + n\pi/2 \)

Key Concepts

Trigonometric Equations

Trigonometric equations are mathematical expressions that involve trigonometric functions such as sine, cosine, tangent, and their reciprocals. These equations can be solved by using fundamental identities and understanding the periodic nature of trig functions. It's essential to remember that solutions to trigonometric equations are often periodic, meaning they repeat at regular intervals. This periodicity is due to the repeating patterns of the trig functions over their respective periods. When solving a trigonometric equation, the goal is typically to isolate the variable, often an angle, to find all possible solutions within a given interval. Many trigonometric equations can have multiple solutions, and those solutions can usually be represented in terms of a general solution that captures all possible instances of the angle satisfying the equation.

Cosine

Cosine is one of the primary trigonometric functions, often abbreviated as cos. In terms of a right triangle, it represents the ratio of the length of the adjacent side to the hypotenuse of the triangle. Mathematically, it's expressed as:\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]The cosine function is important in solving trigonometric equations because it is periodic with a period of \(2\pi\) radians, meaning that \( \cos(\theta) = \cos(\theta + 2k\pi) \) for any integer \(k\).Understanding where the cosine function takes specific values, like \(1/2\), helps to identify the solutions to equations. In the unit circle, cosine corresponds to the x-coordinate of a point on the circle, and one often memorizes the angles where cosine assumes familiar values, such as 0, \(\pm 1/2\), and \(\pm 1\). These special angles are crucial in simplifying and solving trigonometric problems.

Secant

The secant function, represented as \(\sec\), is the reciprocal of the cosine function. It's defined as:\[ \sec(x) = \frac{1}{\cos(x)} \]Because it's the reciprocal, understanding cosine makes secant more intuitive. Where cosine can be zero, the secant function becomes undefined, which typically leads to vertical asymptotes on its graph.Secant is often used in solving trigonometric equations when it's advantageous to manipulate the equation into a form that involves only the cosine function. This simplification helps find solutions by converting the problem to an equation with known values or identities. Transforming \(\sec(4x) = 2\) into \(\cos(4x) = \frac{1}{2}\) allows us to find solutions through known cosine values.

Angle Conversion

In solving trigonometric equations, understanding and converting angles between different forms or units is pivotal. Trigonometric solutions can be represented in degrees or radians, and in mathematical contexts, radians are generally preferred.Knowing the conversions between radians and degrees is crucial. For instance, \(\pi\) radians equals 180 degrees, so \(\frac{\pi}{3}\) radians is equivalent to 60 degrees. Working with radians simplifies many equations due to its natural fit with mathematical constants like \(\pi\). When determining specific solutions or general solutions involving intervals, converting angles helps to comfortably express those intervals and ensure that solutions cover all possible angles within the desired range. For problem-solving, convert the specific angles used in solutions back to degrees if needed, to check against known angle values, simplifying interpretation and verification of results. Understanding angle conversion helps ensure that all potential solutions to a trigonometric equation are properly accounted for in various contexts.