Problem 87 - Precalculus : Building Concepts and Connections

Question

Find the exact value of each expression without using a calculator. $$\csc \frac{\pi}{2}-4 \cot \frac{\pi}{2}$$

Step-by-Step Solution

Answer

The exact value of the expression is 1

Step 1: Find the value of cosecant

First, we need to find the value of the cosecant function at the angle of \(\frac{\pi}{2}\) radians. Remember that cosecant is the reciprocal of the sine function. Hence, \(\csc \frac{\pi}{2}= \frac{1}{\sin \frac{\pi}{2}}\). The sine of \(\frac{\pi}{2}\) is 1, therefore, \(\csc \frac{\pi}{2}= 1\)

Step 2: Find the value of cotangent

Secondly, we have to get the value of the cotangent function at the angle of \(\frac{\pi}{2}\) radians. Cotangent is defined as the cosine of an angle divided by the sine of that angle. Therefore, \(\cot \frac{\pi}{2}= \frac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}}\). The cosine of \(\frac{\pi}{2}\) is 0, so \(\cot \frac{\pi}{2}= \frac{0}{1} = 0\)

Step 3: Substitute the values back into the expression

Next, we substitute the values of the cosecant and cotangent functions back into the original equation. Thus, we have \(1 - 4 \times 0\)

Step 4: Carry out the subtraction

Perform the subtraction to get the final answer. Therefore, \(1 - 4 \times 0 = 1\)

Key Concepts

Cosecant

The cosecant function, abbreviated as \( \csc \), is one of the six main trigonometric functions and is particularly useful in right triangle problems and wave patterns analysis. The cosecant is the reciprocal of the sine function. This means \( \csc \theta = \frac{1}{\sin \theta} \). When thinking about reciprocal relationships, it's important to understand that high sine values produce low cosecant values, and vice versa.

To solve for \( \csc \frac{\pi}{2} \), recall that the sine of \( \frac{\pi}{2} \) radians, which is equivalent to 90 degrees, is 1. Therefore, the reciprocal or cosecant of \( \frac{\pi}{2} \) is simply

• \( \csc \frac{\pi}{2} = \frac{1}{\sin \frac{\pi}{2}} = 1 \)

This particular value is intuitive since the sine of 90 degrees is the length of the hypotenuse in a unit circle, aligning perfectly with its diameter.

Cotangent

The cotangent function, denoted as \( \cot \), is another pivotal function in trigonometry. The cotangent is defined as the reciprocal of the tangent function. Since tangent is given by \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), it subsequently follows that the cotangent is \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). It can be thought of as how steep the slope of an angle is inside a right-angled triangle.

Considering \( \cot \frac{\pi}{2} \), we use our understanding that \( \frac{\pi}{2} \) (or 90 degrees) has a cosine of 0 and a sine of 1. Thus, the cotangent becomes:

• \( \cot \frac{\pi}{2} = \frac{\cos \frac{\pi}{2}}{\sin \frac{\pi}{2}} = \frac{0}{1} = 0 \)

Such a zero result is expected, as the tangent turns vertical (undefined) at this point, which leads to the cotangent at \( \frac{\pi}{2} \) being zero.

Radians

Radians are a way of measuring angles based on the length of the arc of a circle. One radian is the angle formed when the arc length is equal to the radius of the circle. This measurement is fundamental in math because it connects angles to the circumference of the circle directly, making it a "natural" angle measurement.

In the problem, you're working with \( \frac{\pi}{2} \) radians, which is equivalent to 90 degrees. The conversion from degrees to radians is accomplished using the proportion that \( 180 \) degrees equals \( \pi \) radians. Thus, by dividing both sides by 2, you find \( 90 \) degrees equals \( \frac{\pi}{2} \) radians.

Working with radians can seem abstract at first, but they simplify many mathematical formulas, especially in calculus and advanced trigonometry. So, embracing radians helps unlock deeper mathematical understanding and ensures precision in calculations.