Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-38: Find all solutions of the equation. $$ \cos \theta=\frac{1}{\sec \theta} $$

Step-by-Step Solution

Answer

The solutions are all \( \theta \) except \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

Step 1: Understand the Given Equation

We are given the equation \( \cos \theta = \frac{1}{\sec \theta} \). Recall the trigonometric identity that \( \sec \theta = \frac{1}{\cos \theta} \). Substitute this identity into the given equation to simplify.

Step 2: Simplify the Equation Using Identities

Substitute the identity \( \sec \theta = \frac{1}{\cos \theta} \) into the equation: \[ \cos \theta = \frac{1}{\frac{1}{\cos \theta}} \] This simplifies to \( \cos \theta = \cos \theta \), confirming the equation is true for all values where \( \cos \theta \) is defined.

Step 3: Determine the Solutions

Since \( \cos \theta = \cos \theta \) is an identity that always holds true whenever \( \cos \theta \) is defined, the equation holds for all values of \( \theta \) with the exception of points where \( \cos \theta \) is undefined. The cosine function is undefined when \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that appears often in mathematics, especially in the context of right-angle triangles and periodic phenomena like waves. The function is denoted as \( \cos \theta \), where \( \theta \) is the angle in radians.
In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. This can be mathematically expressed as:

• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

The cosine function is periodic with a period of \( 2\pi \). This means it repeats its values every \( 2\pi \) radians. The range of the cosine function, which is the set of possible output values, is from -1 to 1. The function reaches a maximum value of 1 and a minimum of -1.
Understanding this range is crucial when solving equations like \( \cos \theta = \frac{1}{\sec \theta} \), where certain values of \( \theta \) might make \( \cos \theta \) undefined, such as at specific points noted in the solution.

Secant Function

The secant function is another important trigonometric function related to the cosine function. It is denoted as \( \sec \theta \) and is defined as the reciprocal of the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \)

The secant function is undefined whenever the cosine function is zero since division by zero is not possible. This occurs at angles \( \theta = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
The secant function has a period of \( 2\pi \), just like the cosine function. However, unlike the cosine function, the secant function's range is \((-\infty, -1] \cup [1, \infty)\). It never takes values between -1 and 1, reflecting its origin as the reciprocal of the cosine function.
When solving equations involving the secant function, it's important to consider where the function is undefined, as these points could lead to divisions by zero, affecting the solutions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within the domain of the functions involved. These identities are crucial for simplifying and solving trigonometric equations.
One such identity used in the exercise is:

• \( \sec \theta = \frac{1}{\cos \theta} \)

By substituting this identity, we transformed the original equation \( \cos \theta = \frac{1}{\sec \theta} \) into \( \cos \theta = \cos \theta \), a tautology indicating that the equation is true whenever the functions involved are defined.
When applying trigonometric identities, one helpful strategy is to replace complex expressions with simpler equivalent forms. This can often reveal solutions or simplify the problem at hand.
Additionally, understanding identities like the Pythagorean identity, angle sum and difference, and double angle formulas further aids in comprehending more complex trigonometric relationships and solving equations efficiently.