Problem 120 - A Graphical Approach to Precalculus with Limits

Question

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\csc \theta=2,\) given that \(\theta\) is in quadrant II

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: Understand the Reciprocal Relationship

The cosecant function, \(\csc \theta\), is the reciprocal of the sine function. Given \(\csc \theta = 2\), this means \(\sin \theta = \frac{1}{2}\). Since \(\theta\) is in quadrant II, \(\sin \theta\) is positive.

Step 2: Identify the Angle in Quadrant II

In quadrant II, the angle \(\theta\) that has a sine of \(\frac{1}{2}\) is \(\theta = 150^\circ\) (or \(\theta = \frac{5\pi}{6}\) radians). The reference angle is \(30^\circ\) or \(\frac{\pi}{6}\) radians.

Step 3: Calculate \(\cos \theta\) Using Pythagorean Identity

Use the identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\sin \theta = \frac{1}{2}\) to find \(\cos \theta\):\[\cos^2 \theta = 1 - \left(\frac{1}{2}\right)^2 = \frac{3}{4}\]Since \(\theta\) is in quadrant II, \(\cos \theta\) is negative, so \(\cos \theta = -\frac{\sqrt{3}}{2}\).

Step 4: Determine \(\tan \theta\)

The tangent of an angle is the ratio of \(\sin \theta\) to \(\cos \theta\). Calculate \(\tan \theta\) as:\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{2}}{ -\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}\]This simplifies to \(\tan \theta = -\frac{\sqrt{3}}{3}\).

Step 5: Find \(\sec \theta\) and \(\cot \theta\)

The secant function is the reciprocal of the cosine function: \(\sec \theta = -\frac{2}{\sqrt{3}}\), which simplifies to \(-\frac{2\sqrt{3}}{3}\). The cotangent function is the reciprocal of the tangent function: \(\cot \theta = -\sqrt{3}\).

Step 6: Summarize Trigonometric Function Values

For \(\theta\) in quadrant II with \(\csc \theta = 2\):- \(\sin \theta = \frac{1}{2}\)- \(\cos \theta = -\frac{\sqrt{3}}{2}\)- \(\tan \theta = -\frac{\sqrt{3}}{3}\)- \(\sec \theta = -\frac{2\sqrt{3}}{3}\)- \(\cot \theta = -\sqrt{3}\).

Key Concepts

Cosecant

The cosecant function, denoted as \( \csc \theta \), is an important trigonometric function that serves as the reciprocal of the sine function. In simple terms, this means that \( \csc \theta = \frac{1}{\sin \theta} \). Cosecant is less commonly used than sine but is equally crucial in trigonometric calculations.

• For example, if you know \( \csc \theta = 2 \), you can immediately find \( \sin \theta \) by taking the reciprocal, yielding \( \sin \theta = \frac{1}{2} \).
• This reciprocal relationship allows you to quickly switch between the two functions, providing flexibility in solving trigonometric problems.

Understanding how to find sine from cosecant, especially in assignments, is foundational for solving more complex problems relating to angles and their trigonometric values.

Sine

The sine function, represented by \( \sin \theta \), is one of the primary trigonometric functions and is crucial for understanding the properties of angles, especially in right triangles. It is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.

• In this specific exercise, we are given that \( \sin \theta = \frac{1}{2} \).
• Sine values can be positive or negative, depending on the quadrant in which the angle \( \theta \) is located.
• In quadrant II, \( \sin \) is positive, which is a characteristic feature of angles in this part of the unit circle.

Recognizing such properties helps in determining the sign of other trigonometric functions for angles in different quadrants.

Quadrant II

In trigonometry, the coordinate plane is divided into four quadrants, and understanding which quadrant an angle lies in is critical for determining the signs of trigonometric functions.

• Quadrant II is where angles range from \( 90^\circ \) to \( 180^\circ \) (or \( \frac{\pi}{2} \) to \( \pi \) radians).
• In this quadrant, the sine function is positive, while the cosine and tangent functions are negative.
• For our problem, we found \( \theta = 150^\circ \) or \( \frac{5\pi}{6} \) radians, which indeed lies in this quadrant.

Being able to identify the correct quadrant for an angle allows you to assign the correct sign to the trigonometric functions, simplifying the solving process.

Angle Calculation

Calculating the precise value of an angle based on a given trigonometric function value involves understanding inverse trigonometric functions and the properties of the unit circle.

• For the task at hand, starting with \( \csc \theta = 2 \), we get \( \sin \theta = \frac{1}{2} \).
• Angles that correspond to \( \sin \theta = \frac{1}{2} \) include \( 30^\circ \) (or \( \frac{\pi}{6} \) radians) as the reference angle.
• Since \( \theta \) is in quadrant II, the actual angle is \( 180^\circ - 30^\circ = 150^\circ \) (or \( \pi - \frac{\pi}{6} = \frac{5\pi}{6} \) radians).

These steps, leveraging reference angles and knowledge of quadrants, allow students to confidently determine angle measures when given specific trigonometric function values.