Problem 47 - Algebra and Trigonometry

Question

43–50 Find the values of the trigonometric functions of \(\theta\) from the information given. \(\csc \theta=2, \quad \theta\) in quadrant I

Step-by-Step Solution

Answer

\( \sin(\theta) = \frac{1}{2}, \cos(\theta) = \frac{\sqrt{3}}{2}, \tan(\theta) = \frac{\sqrt{3}}{3}, \sec(\theta) = \frac{2\sqrt{3}}{3}, \cot(\theta) = \sqrt{3} \)

Step 1: Understanding Cosecant

Given the cosecant function, \ \ \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we have that \( \csc(\theta) = 2 \). Hence, \( \sin(\theta) = \frac{1}{2} \).

Step 2: Identifying the Sine Function

Since \( \csc(\theta) = 2 \), we can find \( \sin(\theta) \) using the reciprocal: \[ \sin(\theta) = \frac{1}{2} \] Since \( \theta \) is in Quadrant I, all trigonometric functions are positive.

Step 3: Finding Cosine Function

Use the Pythagorean identity: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] Substitute \( \sin(\theta) = \frac{1}{2} \): \[ \left( \frac{1}{2} \right)^2 + \cos^2(\theta) = 1 \] \[ \frac{1}{4} + \cos^2(\theta) = 1 \] \[ \cos^2(\theta) = \frac{3}{4} \] \[ \cos(\theta) = \frac{\sqrt{3}}{2} \] Choose the positive value because \( \theta \) is in Quadrant I.

Step 4: Finding Tangent Function

Since \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \): \[ \tan(\theta) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]

Step 5: Finding Secant and Cotangent Function

Use the reciprocal identities for secant and cotangent: \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \] \[ \cot(\theta) = \frac{1}{\tan(\theta)} = \sqrt{3} \]

Key Concepts

Cosecant

Understanding the cosecant function is crucial when exploring trigonometric identities. The cosecant, denoted as \( \csc \theta \), is the reciprocal of the **sine function**. This means it is defined as follows:

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)

In a given exercise, if you know that \( \csc \theta = 2 \), then you can find \( \sin \theta \) by taking the reciprocal:

• \( \sin(\theta) = \frac{1}{2} \)

Remember that in the first quadrant of the unit circle, where \( \theta \) is positioned in this exercise, all trigonometric functions, including sine, are positive. This aspect ensures our calculations for trigonometric values remain straightforward and without the need for additional sign considerations.

Sine

The sine function is one of the fundamental trigonometric functions, and it is often used to explore solutions related to angles and triangles within the unit circle setting. In practical scenarios of trigonometry, the sine of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.

• Mathematically, it's expressed as \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

In the exercise provided, where \( \csc \theta = 2 \), applying the concept of sine gives us:

• \( \sin(\theta) = \frac{1}{2} \)

Being in the first quadrant, all trigonometric functions, including sine, maintain a positive value, streamlining the calculation process. This situation conveniently sets the stage for using other identities like the Pythagorean identity in subsequent calculations.

Pythagorean Identity

The Pythagorean identity is a pivotal equation in trigonometry, connecting sine and cosine through the geometry of the unit circle. Stated clearly, the identity is expressed as:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)

This formula provides a way to determine one trigonometric value if another is known. For instance, with the sine value identified as \( \sin(\theta) = \frac{1}{2} \), we can isolate \( \cos(\theta) \) by plugging this sine value into the Pythagorean identity:

• \( \left( \frac{1}{2} \right)^2 + \cos^2(\theta) = 1 \)
• \( \frac{1}{4} + \cos^2(\theta) = 1 \)
• \( \cos^2(\theta) = \frac{3}{4} \)
• \( \cos(\theta) = \frac{\sqrt{3}}{2} \)

Selecting the positive square root respects the fact that \( \theta \) is in the first quadrant, ensuring that we work with a positive cosine value. This calculation not only confirms the cosine value but also leads into determining other related trigonometric functions.