Problem 12 - Algebra 2 and Trigonometry

Question

For each function value, write the value or tell why it is undefined. Do not use a calculator. \(\csc \frac{7 \pi}{2}\)

Step-by-Step Solution

Answer

The value is 1.

Step 1: Understand the Cosecant Function

The cosecant function is the reciprocal of the sine function, defined as \(\csc(\theta) = \frac{1}{\sin(\theta)}\). Before determining \(\csc \frac{7 \pi}{2}\), we need to find \(\sin \frac{7 \pi}{2}\).

Step 2: Simplify the Angle

The angle \(\frac{7 \pi}{2}\) is greater than \(2\pi\), which means it is coterminal with an angle between 0 and \(2\pi\). To find this, subtract multiples of \(2\pi\) until the angle falls within the standard range:

Step 3: Subtract to Find a Coterminal Angle

Subtract \(2\pi\) from \(\frac{7\pi}{2}\) to simplify it:\[\frac{7\pi}{2} - 3\pi = \frac{7\pi - 6\pi}{2} = \frac{\pi}{2}.\]So, \(\frac{7\pi}{2}\) is coterminal with \(\frac{\pi}{2}\).

Step 4: Find the Sine of the Simplified Angle

Since \(\frac{\pi}{2}\) is a familiar angle on the unit circle, recall that \(\sin \frac{\pi}{2} = 1\).

Step 5: Calculate the Cosecant

Now, use the value of \(\sin \frac{\pi}{2}\) to find the cosecant:\[\csc \frac{\pi}{2} = \frac{1}{\sin \frac{\pi}{2}} = \frac{1}{1} = 1.\]Thus, \(\csc \frac{7\pi}{2} = 1\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that simplifies understanding angles and trigonometric functions. It is a circle with a radius of one unit, centered at the origin of a coordinate plane. This circle allows us to define sine, cosine, and other trigonometric functions for all angles in a straightforward way.

In the unit circle, any angle between 0 and \(2\pi\) radians or 0 and 360 degrees, corresponds to a point on the circumference of the circle. Here, the x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine.

• For example, at \(\frac{\pi}{2}\) radians (or 90 degrees), the point is \((0, 1)\), meaning \(\cos \frac{\pi}{2} = 0\) and \(\sin \frac{\pi}{2} = 1\).
• The unit circle also helps in understanding angles greater than \(2\pi\) by using the concept of revolutions. The angle \(\frac{7\pi}{2}\) is one such angle, which after subtracting multiples of \(2\pi\), becomes \(\frac{\pi}{2}\), showing its coterminal nature.

Using the unit circle simplifies finding values of trigonometric functions without a calculator, aiding visualization and comprehension.

Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides, essentially pointing in the same direction, even though they may have different measures. This occurs because angles can make multiple rotations around the circle and still end up in the same position.

To find coterminal angles, we continuously add or subtract \(2\pi\) radians (or 360 degrees) from the given angle until it falls within the desired range, usually between 0 and \(2\pi\). For instance:

• The angle \(\frac{7\pi}{2}\) is coterminal with \(\frac{\pi}{2}\) because \(\frac{7\pi}{2} - 3\pi = \frac{\pi}{2}\).
• This means \(\frac{7\pi}{2}\) and \(\frac{\pi}{2}\) effectively represent the same position on the unit circle.

Recognizing coterminal angles is particularly useful in simplifying calculations of trigonometric functions, making such operations more manageable and understanding their periodicity.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the main trigonometric functions by taking their reciprocals. The primary ones include cosecant, secant, and cotangent, which correspond to sine, cosine, and tangent, respectively.

The cosecant function is the reciprocal of the sine function, expressed as \(\csc(\theta) = \frac{1}{\sin(\theta)}\). This means wherever \(\sin(\theta) = 0\), \(\csc(\theta)\) is undefined, since division by zero is impossible.

• For example, evaluating \(\csc \frac{7\pi}{2}\) requires first finding \(\sin \frac{7\pi}{2}\), which is \(1\) (from its coterminal angle \(\frac{\pi}{2}\)).
  Thus, \(\csc \frac{7\pi}{2} = \frac{1}{1} = 1\).
• It's crucial to ensure the sine value isn't zero when calculating the cosecant, secant, or cotangent functions to prevent undefined results.

Understanding these reciprocal functions expands the range of problems one can solve in trigonometry and enhances comprehension of their roles in different scenarios.