Problem 14 - Algebra 2

Question

Find the value of each expression. \(\csc \theta,\) if \(\cos \theta=-\frac{3}{5} ; 90^{\circ}<\theta<180^{\circ}\)

Step-by-Step Solution

Answer

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

Step 1: Recognize the quadrant

We need to find the quadrant where \(90^{\circ}<\theta<180^{\circ}\). This range corresponds to the second quadrant.

Step 2: Determine the sine sign

In the second quadrant, sine is positive. Therefore, \(\sin \theta\) will be a positive value.

Step 3: Use Pythagorean identity

Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = -\frac{3}{5}\) into the identity:\[\sin^2 \theta + \left(-\frac{3}{5}\right)^2 = 1\]\[\sin^2 \theta + \frac{9}{25} = 1\]Subtract \(\frac{9}{25}\) from both sides:\[\sin^2 \theta = 1 - \frac{9}{25}\]\[\sin^2 \theta = \frac{25}{25} - \frac{9}{25}\]\[\sin^2 \theta = \frac{16}{25}\]

Step 4: Solve for \(\sin \theta\)

We take the square root of both sides to find \(\sin \theta\):\[\sin \theta = \pm \sqrt{\frac{16}{25}}\]\[\sin \theta = \pm \frac{4}{5}\]Since we are in the second quadrant, \(\sin \theta\) is positive, so \(\sin \theta = \frac{4}{5}\).

Step 5: Calculate \(\csc \theta\)

The cosecant function is the reciprocal of the sine function. Therefore, \(\csc \theta = \frac{1}{\sin \theta}\):\[\csc \theta = \frac{1}{\frac{4}{5}}\]\[\csc \theta = \frac{5}{4}\]

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of the sine and cosine of an angle. It is expressed as:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity is crucial as it helps in finding one trigonometric function given another.
For instance, if you know the cosine of an angle, you can easily find the sine using this equation. In this specific exercise, knowing that \(\cos \theta = -\frac{3}{5}\), we used the Pythagorean Identity to find \(\sin \theta\).
Starting with solving for \(\sin^2 \theta\), we substituted the given cosine value into the identity:
\[\sin^2 \theta + \left(-\frac{3}{5}\right)^2 = 1\]This simplified to finding \(\sin^2 \theta\) as:\
\[\sin^2 \theta = \frac{16}{25}\]
Then, taking the square root determined that \(\sin \theta = \pm \frac{4}{5}\). By considering the quadrant, we concluded which sign to use.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, each corresponding to a different range of angles:

• Quadrant I: \(0^\circ\) to \(90^\circ\)
• Quadrant II: \(90^\circ\) to \(180^\circ\)
• Quadrant III: \(180^\circ\) to \(270^\circ\)
• Quadrant IV: \(270^\circ\) to \(360^\circ\)

In these quadrants, the trigonometric functions vary in terms of their signs:

• Quadrant I: All functions positive
• Quadrant II: Sine positive; cosine and tangent negative
• Quadrant III: Tangent positive; sine and cosine negative
• Quadrant IV: Cosine positive; sine and tangent negative

This exercise placed \(\theta\) in the second quadrant, where sine is positive.
Recognizing the quadrant helps determine the correct sign for trigonometric values and ensures accurate calculation of functions such as \(\sin \theta\) and \(\csc \theta\).

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions primarily used to describe the ratio of sides in a right triangle or on the unit circle.
The sine function, represented as \(\sin \theta\), gives the ratio of the "opposite side" to the "hypotenuse" in a right triangle. On the unit circle, it corresponds to the y-coordinate.
The cosine function, or \(\cos \theta\), provides the ratio of the "adjacent side" to the "hypotenuse" and matches the x-coordinate on the unit circle.
In this particular exercise, \(\cos \theta\) was given as \(-\frac{3}{5}\), revealing \(\theta\) situated on the negative x-axis in Quadrant II. Meanwhile, \(\sin \theta\) was calculated through the identity and associated quadrant properties, resulting in \(\sin \theta = \frac{4}{5}\). Understanding these functions' roles on the unit circle allows you to efficiently find one when the other is known and handle their reciprocals, like \(\csc \theta = \frac{1}{\sin \theta}\).