Problem 15 - Algebra 2

Question

Find the value of each expression. \(\cos \theta,\) if \(\sec \theta=\frac{5}{3} ; 270^{\circ}<\theta<360^{\circ}\)

Step-by-Step Solution

Answer

\( \cos \theta = \frac{3}{5} \) in the given interval.

Step 1: Understand Secant and Cosine Relationship

The secant function is the reciprocal of the cosine function. Thus, if \( \sec \theta = \frac{5}{3} \), then \( \cos \theta = \frac{1}{\sec \theta} \). Therefore, we have \( \cos \theta = \frac{3}{5} \).

Step 2: Determine the Quadrant

The given range for the angle \( \theta \) is \( 270^{\circ} < \theta < 360^{\circ} \), which corresponds to the fourth quadrant of the unit circle. In the fourth quadrant, the cosine function is positive.

Step 3: Apply the Quadrant Rule for Cosine

Since \( \theta \) is in the fourth quadrant, where cosine values are positive, the value of \( \cos \theta \) does not change its sign. Therefore, \( \cos \theta = \frac{3}{5} \).

Key Concepts

Secant

The secant function, denoted as \( \sec \theta \), is a trigonometric function closely related to the cosine function. The secant of an angle \( \theta \) is defined as the reciprocal of the cosine of \( \theta \), which mathematically means \( \sec \theta = \frac{1}{\cos \theta} \). This relationship helps us find the value of the cosine function if the secant is given. For example, if \( \sec \theta = \frac{5}{3} \), then the cosine function corresponding to this secant value would be \( \cos \theta = \frac{3}{5} \).
Understanding the secant function and its relationship with the cosine is key when solving trigonometric problems. It's particularly useful when specific angle measures are involved, allowing for easy transitions between the two functions.

Cosine Function

The cosine function evaluates the x-coordinate of a point on the unit circle associated with a given angle \( \theta \). Mathematically, it is expressed as \( \cos \theta \).
The cosine function ranges from -1 to 1, corresponding to the full sweep around the circle. This function is periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians, meaning it repeats its values every full rotation of the circle. In various trigonometric contexts such as right triangles or circular motion, cosine values help determine the horizontal component or adjacent side of a triangle relative to the angle.

• In different quadrants, the sign of cosine can vary: it's positive in the first and fourth quadrants and negative in the second and third.
• This alteration is pivotal when solving problems that involve specific angle intervals or ranges.

Hence, cosine is a fundamental function extensively used in geometry and physics, especially in wave and oscillatory motion.

Unit Circle

The unit circle is a critical concept in trigonometry, serving as a fundamental reference for all trigonometric functions. This circle is a simple circle centered at the origin of a coordinate plane with a radius of 1 unit.
The unit circle helps visualize functions like sine, cosine, and tangent via angles measured in radians or degrees. A point on the unit circle, denoted by \((x, y)\), corresponds to \((\cos \theta, \sin \theta)\). The x-coordinate of this point is \( \cos \theta \), and the y-coordinate is \( \sin \theta \).

• This concept is vital as it visually demonstrates how the trigonometric functions behave with respect to different angle measures.
• It also clarifies the periodic nature of these functions, allowing learners to comprehend their repetitive patterns around the circle.

The unit circle serves as an educational tool making complex trigonometric identities and relationships more intuitive.

Quadrant

In trigonometry, a quadrant refers to one of the four sections of a coordinate plane. Each quadrant represents a range of angle measures and impacts the sign of trigonometric functions. The quadrants are numbered counterclockwise:

• First Quadrant: \(0^{\circ} \) to \(90^{\circ} \) where all trigonometric functions are positive.
• Second Quadrant: \(90^{\circ} \) to \(180^{\circ} \) where sine is positive, but cosine and tangent are negative.
• Third Quadrant: \(180^{\circ} \) to \(270^{\circ} \) where tangent is positive, but sine and cosine are negative.
• Fourth Quadrant: \(270^{\circ} \) to \(360^{\circ} \) where cosine is positive, but sine and tangent are negative.

Understanding quadrants is crucial in solving trigonometric equations as it guides us to correctly apply the sign to the function values. This knowledge is especially important in problems, like the one here, where you're given a range of angles and need to determine the appropriate sign for a trigonometric function.