Problem 76 - Algebra 2

Question

For Exercises \(75-78,\) suppose \(\cos \theta=\frac{3}{5}\) and \(\sin \theta>0\) . Enter each answer as a fraction. What is \(\sec \theta ?\)

Step-by-Step Solution

Answer

\(\sec \theta = \frac{5}{3}\)

Step 1: Understand the problem

We are given that \(\cos \theta =\frac{3}{5}\) and asked find the value of \(\sec \theta\). Wei ndentify that \(\sec \theta\) is the reciprocal of \(\cos \theta\).

Step 2: Take the reciprocal

Find the reciprocal of \(\cos \theta\) to find \(\sec \theta\). In other words, \[\sec \theta = \frac{1}{\cos \theta} \]

Step 3: Substitute the given value into the equation

Substitute \(\cos \theta =\frac{3}{5}\) into the equation to find \(\sec \theta = \frac{1}{\cos \theta}\) equals \[\frac{1}{3/5} = \frac{5}{3}\]

Key Concepts

Cosine

Cosine is one of the fundamental trigonometric functions, often abbreviated as "cos." It is used to describe the relationship between the angle of a right triangle and the lengths of its adjacent side and hypotenuse. In mathematical terms, the cosine of an angle \( \theta \) is defined as \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \] Here are some key points about the cosine function:

• The cosine function is periodic with a period of \( 2\pi \), meaning that its values repeat every \( 360^\circ\) or \( 2\pi \) radians.
• Cos \( \theta \) ranges between -1 and 1 for any angle \( \theta \).
• In the unit circle, \( \cos \theta \) represents the x-coordinate of a point where the terminal side of the angle intersects the circle.

For the given exercise, you have \( \cos \theta = \frac{3}{5} \). This indicates that if \( \theta \) were an angle in a right triangle, the side adjacent to \( \theta \) is 3 units long, while the hypotenuse is 5 units long. Understanding cosine is crucial because it serves as the basis for determining other trigonometric functions, like secant, as seen in the exercise.

Secant

The secant function, denoted as \( \sec \theta \), is another essential trigonometric function. It is closely related to the cosine function, as it is specifically defined as the reciprocal of the cosine function. In simple terms, secant represents how many times the cosine value fits into 1. The formula for secant is:\[ \sec \theta = \frac{1}{\cos \theta} \] Some key features of the secant function include:

• Like cosine, secant is also a periodic function with a period of \( 2\pi \).
• Since it is the reciprocal of cosine, \( \sec \theta \) is undefined wherever \( \cos \theta = 0 \), as division by zero is not possible.
• While \( \cos \theta \) ranges from -1 to 1, \( \sec \theta \) can have values greater than 1 or less than -1, depending on \( \theta \).

In the exercise, you solve for \( \sec \theta \) by taking the reciprocal of \( \cos \theta = \frac{3}{5} \), resulting in:\[ \sec \theta = \frac{1}{3/5} = \frac{5}{3} \] This calculation shows how secant expands beyond the cosine value, giving you a greater understanding of relationships in trigonometry.

Reciprocal Identities

Reciprocal identities are a key concept in trigonometry, helping create connections between trigonometric functions. These identities help understand the reciprocal relationships among the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. Understanding these identities aids in solving various trigonometric problems and simplifying expressions.Here's a quick overview of reciprocal identities:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These identities illustrate the inverse relationships that exist between pairs of trigonometric functions. For instance, as shown in the exercise, knowing \( \cos \theta = \frac{3}{5} \) directly allows us to find \( \sec \theta = \frac{1}{\cos \theta} = \frac{5}{3} \). Reciprocal identities play an essential role in simplifying expressions and solving equations in trigonometry. By understanding and applying these identities, one can seamlessly switch between functions, clarify complex trigonometric relationships, and solve problems like those encountered in the exercise.