Problem 9 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{llll}{\text { (a) } \sin \frac{7 \pi}{3}} & {\text { (b) } \csc \frac{7 \pi}{3}} & {\text { (c) } \cot \frac{7 \pi}{3}}\end{array} $$

Step-by-Step Solution

Answer

\(\sin\left(\frac{7\pi}{3}\right) = \frac{\sqrt{3}}{2}\), \(\csc\left(\frac{7\pi}{3}\right) = \frac{2\sqrt{3}}{3}\), \(\cot\left(\frac{7\pi}{3}\right) = \frac{\sqrt{3}}{3}\).

Step 1: Simplify the Angle

Both the sine, cosecant, and cotangent functions have a periodicity of \(2\pi\). To simplify the angle, \(\frac{7\pi}{3}\), we subtract multiples of \(2\pi\) until the angle lies within the range \([0, 2\pi)\). Divide \(\frac{7\pi}{3}\) by \(\frac{6\pi}{3}\), the equivalent of \(2\pi\), until the angle is simplified:\[\frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}.\] Now, our angle is \(\frac{\pi}{3}\).

Step 2: Evaluate \(\sin\left(\frac{\pi}{3}\right)\)

The sine of \(\frac{\pi}{3}\) is a commonly known value from the unit circle or from special triangles. \[\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}.\]

Step 3: Evaluate \(\csc\left(\frac{\pi}{3}\right)\)

The cosecant function is the reciprocal of the sine function. If \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), then:\[\csc\left(\frac{\pi}{3}\right) = \frac{1}{\sin\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}.\]

Step 4: Evaluate \(\cot\left(\frac{\pi}{3}\right)\)

The cotangent function is the reciprocal of the tangent function, i.e., \(\cot(\theta) = \frac{1}{\tan(\theta)}\). In this case,\[\tan\left(\frac{\pi}{3}\right) = \sqrt{3},\]so,\[\cot\left(\frac{\pi}{3}\right) = \frac{1}{\tan\left(\frac{\pi}{3}\right)} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}.\]

Key Concepts

Periodic Functions

Periodic functions are a fascinating aspect of trigonometry. They repeat their values at regular intervals. For trigonometric functions like sine, cosecant, cotangent, and tangent, this interval is usually every \(2\pi\) radians.
This means that if you have an angle, say \(\theta\), you can add or subtract \(2\pi\) (or any multiple of \(2\pi\)) and the sine, cosecant, tangent, or cotangent will yield the same value.

• The periodicity of \(2\pi\) helps to simplify problems by reducing angles that are "large" to more manageable angles that lie within a single period, usually between \(0\) and \(2\pi\).
• For example, if you start with an angle of \(\frac{7\pi}{3}\), after reducing it by \(2\pi\), you get \(\frac{\pi}{3}\). Even though \(\frac{7\pi}{3}\) is outside the standard \([0, 2\pi)\) range, it has an equivalent angle \(\frac{\pi}{3}\) inside this range.

Understanding periodicity allows you to work with values considered standard or commonly known, making calculations easier and faster. It’s akin to finding patterns in a cycle, knowing that every complete cycle brings you to familiar ground.

Sine and Cosecant

The sine function is foundational in trigonometry, representing the y-coordinate of a point on the unit circle at a given angle. It answers the question, "What is the vertical position of this point relative to the center of the circle?"
For example, the sine of \(\frac{\pi}{3}\) or 60 degrees is \(\frac{\sqrt{3}}{2}\), a value derived from either the unit circle or the 30-60-90 special triangle.

• The sine function oscillates between \(-1\) and \(1\), repeating its cycle every \(2\pi\) radians.
• The cosecant function, defined as the reciprocal of the sine function, is less frequently used than its counterpart but serves well in certain problems needing the inverse set up.

To find the cosecant of \(\frac{\pi}{3}\), simply take the reciprocal of the sine value: \[\csc\left(\frac{\pi}{3}\right) = \frac{1}{\sin\left(\frac{\pi}{3}\right)} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}.\] The key takeaway is that sine and cosecant are closely linked: sine measures a direct value, while cosecant provides its inverse, expanding the toolbox of solutions in trigonometry.

Cotangent and Tangent

The tangent and cotangent functions offer insights through the ratios they represent. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In the context of the unit circle, \tan(\theta)\ gives you the y-coordinate divided by the x-coordinate for an angle \(\theta\).

• At \(\frac{\pi}{3}\) or 60 degrees, the tangent value is \(\sqrt{3}\).
• Cotangent, the reciprocal of tangent, flips this scenario, providing \[cot\left(\frac{\pi}{3}\right) = \frac{1}{\tan\left(\frac{\pi}{3}\right)} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}.\]

Understanding these functions involves grasping their periodicity and transformations. With tangent and cotangent, noticing how they handle certain sides of a triangle becomes crucial to solving diverse trigonometric problems. Each function’s unique perspective helps determine the unknown aspects related to angles and side lengths in different scenarios. These insights offer a powerful method for approaching and understanding complex trigonometric relationships.