Problem 17 - Algebra 2

Question

Find the value of each expression. \(\csc \theta,\) if \(\cos \theta=-\frac{2}{3} ; 180^{\circ}<\theta<270\)

Step-by-Step Solution

Answer

\( \csc \theta = -\frac{3\sqrt{5}}{5} \)

Step 1: Determine the Quadrant

Given that \(180^{\circ} < \theta < 270^{\circ}\), \(\theta\) is in the third quadrant. In the third quadrant, sine is negative.

Step 2: Use the Pythagorean Identity

We know \( \cos \theta = -\frac{2}{3} \), and in any triangle, \( \sin^2 \theta + \cos^2 \theta = 1 \).Substituting the value of \( \cos \theta \) into this equation:\[ \sin^2 \theta + \left(-\frac{2}{3}\right)^2 = 1 \]\[ \sin^2 \theta + \frac{4}{9} = 1 \]Rearrange to solve for \( \sin^2 \theta \):\[ \sin^2 \theta = 1 - \frac{4}{9} = \frac{5}{9} \].

Step 3: Solve for \( \sin \theta \)

Since \( \theta \) is in the third quadrant where sine is negative, we take the negative square root:\[ \sin \theta = -\sqrt{\frac{5}{9}} = -\frac{\sqrt{5}}{3} \].

Step 4: Find \( \csc \theta \)

The cosecant function is the reciprocal of sine, therefore:\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{\sqrt{5}}{3}} = -\frac{3}{\sqrt{5}} \].To rationalize the denominator, multiply by \( \frac{\sqrt{5}}{\sqrt{5}} \):\[ \csc \theta = -\frac{3\sqrt{5}}{5} \].

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is a fundamental trigonometric function. It is used often in trigonometry to relate an angle in a right triangle to the lengths of its sides. This function is defined as the reciprocal of the sine function, which means that:

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

Like the sine and other reciprocal trigonometric functions, the values of the cosecant function can be negative or positive depending on the angle's quadrant. It is especially handy when angles lead to undefined values for sine, such as when the sine is zero, making the reciprocal undefined. This feature is crucial for solving trigonometric equations, as encountered in the problem above, where we found that \( \csc \theta = -\frac{3\sqrt{5}}{5} \). By understanding the relationship between cosecant and sine, you can tackle similar problems with confidence.

Pythagorean Identity

In trigonometry, the Pythagorean identity is a fundamental relationship amongst the sine, cosine, and tangent functions. One of the most common forms of this identity is:

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

The Pythagorean identity expresses that the square of the sine of an angle plus the square of the cosine of the same angle always equals one. Derived from the Pythagorean theorem, this identity helps in solving trigonometric equations and transforming expressions.
In the provided solution, we used this identity to find \( \sin \theta \) when \( \cos \theta \) was given as \(-\frac{2}{3}\). By rearranging the identity, we were able to determine \( \sin^2 \theta = \frac{5}{9} \) and subsequently the correct sign for \( \sin \theta \) based on the quadrant.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true, regardless of the angle. This powerful concept includes a variety of equations like basic identities (sine, cosine, tangent) and their reciprocals (cosecant, secant, cotangent), as well as compound and double angle identities.

• Reciprocal identities like \( \csc \theta = \frac{1}{\sin \theta} \)
• Pythagorean identities like \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Coffunction identities such as \( \sin(90^\circ - \theta) = \cos \theta \)

These identities are vital tools for simplifying complex trigonometric equations and understanding the relationships between the different functions. In our solution, we mostly focused on the reciprocal identity and the Pythagorean identity to first solve for sine and then for cosecant. Understanding these identities enables solving not just textbook problems but also practical real-world trigonometric challenges.

Third Quadrant

Understanding the behavior of trigonometric functions in different quadrants is essential in trigonometry. The angle \( \theta \) given as between \(180^{\circ}\) and \(270^{\circ}\) places it squarely in the third quadrant of the unit circle. Each quadrant has distinct characteristics:

• First Quadrant: All trigonometric functions are positive.
• Second Quadrant: Sine is positive; others are negative.
• **Third Quadrant**: Tangents and cotangents are positive; sine and cosine are negative.
• Fourth Quadrant: Cosine is positive; others are negative.

In the third quadrant, both sine and cosine are negative. This information was crucial when determining \( \sin \theta \) in our problem, ensuring we chose the negative square root since we knew \( \theta \) was in the third quadrant. This understanding helps to avoid common mistakes when working with angles and improves problem-solving precision.