Problem 7 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value of \( \csc \frac{\pi}{2} \) is 1.

Step 1: Understand the Function

The function given is the cosecant function, which is the reciprocal of the sine function. Mathematically, it is expressed as \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Our goal is to find \( \csc \frac{\pi}{2} \).

Step 2: Calculate the Sine Value

Find the sine of \( \frac{\pi}{2} \). From the unit circle, we know that \( \sin(\frac{\pi}{2}) = 1 \).

Step 3: Calculate Cosecant

Use the definition of cosecant to calculate \( \csc \frac{\pi}{2} \) as the reciprocal of \( \sin \frac{\pi}{2} \). So we have \( \csc \frac{\pi}{2} = \frac{1}{\sin(\frac{\pi}{2})} = \frac{1}{1} = 1 \).

Step 4: Conclusion

The value of \( \csc \frac{\pi}{2} \) is \( 1 \), and it is not undefined because the denominator (\( \sin \frac{\pi}{2} \)) is non-zero.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions in mathematics. It is often abbreviated as "sin" and is used to calculate angles, particularly in right triangles and waves. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse.

For angles measured in radians, the sine function has a periodic pattern, repeating every \[2\pi\] radians. This cyclic behavior is why sine is a crucial function in modeling periodic phenomena such as sound waves, tides, and light waves.

• Range: -1 to 1, as it represents a ratio involving triangle sides.

When dealing with circles, the sine of an angle corresponds to its vertical coordinate on the unit circle at that angle. The sine function is smooth and continuous, allowing it to describe natural waves easily.

Unit Circle

The unit circle is a circle with a radius of 1 unit centered at the origin of a coordinate plane. It is a valuable tool when studying trigonometric functions, particularly sine, cosine, and tangent.

By using radians, the entire circle represents \[2\pi\] in terms of angle, starting from \[0\] radians at the positive x-axis, rotating counterclockwise. Here's how it helps with the sine function:

• The sine of an angle is the y-coordinate of the point on the unit circle corresponding to that angle.
• At \(\frac{\pi}{2}\), the y-coordinate is 1, resulting in \[\sin\left(\frac{\pi}{2}\right) = 1\].

This property makes it simple to determine sine values for key angles, particularly those that lie on the axes of the unit circle. Each quadrant of the circle impacts sine signs and values, enhancing its application in wave function analysis.

Undefined Values

In trigonometry, some function values become undefined when their operations involve division by zero.

For cosecant, which is the reciprocal of the sine function, it becomes undefined where the sine function is zero. This occurs notably:

• At angles like \[0\] and \[\pi\] , where \[\sin(0) = 0\] and \[\sin(\pi) = 0\].

Any point where division by zero might occur results in an undefined expression for the reciprocal functions like cosecant.
For instance, \(\csc(0)\) and \(\csc(\pi)\) are undefined because they involve dividing by zero in their calculation process of \(\frac{1}{\sin(\theta)}\). Understanding these concepts helps avoid mistakes when evaluating these functions in mathematics and physics.