Problem 30 - Algebra 2

Question

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

Step-by-Step Solution

Answer

\(\cos \theta = \frac{4}{5}\)

Step 1: Understanding the range and quadrant

The given condition is that \(270^{\circ} < \theta < 360^{\circ}\), which means \(\theta\) is in the fourth quadrant. In this quadrant, cosine values are positive and sine values are negative.

Step 2: Finding the sine of \(\theta\)

The cosecant function is the reciprocal of the sine function, so \(\csc \theta = -\frac{5}{3}\) implies \(\sin \theta = -\frac{3}{5}\).

Step 3: Using the Pythagorean identity to find cosine

We use the identity \(\cos^2 \theta + \sin^2 \theta = 1\). Substitute \(\sin \theta = -\frac{3}{5}\) into the equation, so: \[\cos^2 \theta + \left(-\frac{3}{5}\right)^2 = 1\]\[\cos^2 \theta + \frac{9}{25} = 1\]Solving for \(\cos^2 \theta\) gives:\[\cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}\]

Step 4: Solving for \(\cos \theta\) and determining the sign

Taking the square root of both sides, we get \(\cos \theta = \pm \frac{4}{5}\). Since \(\theta\) is in the fourth quadrant, where cosine is positive, we have \(\cos \theta = \frac{4}{5}\).

Key Concepts

Cosecant

The cosecant function, written as \( \csc \theta \), is one of the six fundamental trigonometric functions and is particularly known as the reciprocal of the sine function. This can be expressed as \( \csc \theta = \frac{1}{\sin \theta} \). Therefore, if you have a value for \( \csc \theta \), you can easily find \( \sin \theta \) by taking the reciprocal.
In our problem, \( \csc \theta \) is given as \(-\frac{5}{3}\). This translates to \( \sin \theta = -\frac{3}{5} \). It’s important to understand the sign of these values as it relates directly to which trigonometric quadrant \( \theta \) is in. In this case, \( \csc \theta \) being negative indicates \( \sin \theta \) is also negative, occurring in the fourth quadrant.

Cosine

The cosine function, denoted \( \cos \theta \), is another of the primary trigonometric functions. It is crucial when working with right triangles and in the analysis of periodic phenomena. Cosine can be explained simply as the adjacent side over the hypotenuse in a right-angled triangle.
In trigonometric terms, cosine is directly connected with the unit circle. For any angle \( \theta \), \( \cos \theta \) represents the x-coordinate of the endpoint formed by the angle on the unit circle.
Finding \( \cos \theta \) given \( \csc \theta \) and knowing the quadrant is straightforward if you engage the Pythagorean identity. From our steps, we deduced \( \cos \theta = \pm \frac{4}{5} \). Choosing which sign is correct depends on the knowledge of the quadrant; the fourth quadrant is where cosine remains positive.

Pythagorean Identity

The Pythagorean identity is a fundamental identity in trigonometry, expressing the intrinsic relationship between sine and cosine. It's stated as \( \cos^2 \theta + \sin^2 \theta = 1 \). This identity allows us to easily find one trigonometric function given the other.
In our given problem, we know \( \sin \theta = -\frac{3}{5} \). Plugging this into the Pythagorean identity, we set up the equation:

• \( \cos^2 \theta + \left(-\frac{3}{5}\right)^2 = 1 \)
• \( \cos^2 \theta + \frac{9}{25} = 1 \)

Rearranging gives us \( \cos^2 \theta = \frac{16}{25} \).
This step helps us prepare to delve further into finding \( \cos \theta \) itself by taking the square root, keeping quadrant information in mind.

Quadrants in Trigonometry

Trigonometric functions take on different signs depending on which of the four quadrants an angle \( \theta \) is located in. Understanding these quadrants can be crucial in solving any trigonometry problem effortlessly. Here’s a simple breakdown:

• **First Quadrant**: All trigonometric functions are positive.
• **Second Quadrant**: Sine is positive, others negative.
• **Third Quadrant**: Tangent is positive, others negative.
• **Fourth Quadrant**: Cosine is positive, others negative.

Considering these signs helps in understanding the behavior of trigonometric functions based on angle location. In our exercise, since \( \theta \) is in the fourth quadrant, cosine remains positive while sine and tangent are negative, reaffirming that \( \cos \theta = \frac{4}{5} \) in this case.