Problem 28 - Algebra 2

Question

Evaluate each expression to the nearest hundredth. Each angle is given in radians. $$ \csc (-4.5) $$

Step-by-Step Solution

Answer

Make sure your calculator is set to radians mode, then calculate \( \sin(-4.5) \) and take the reciprocal. This is your answer, rounded to the nearest hundredth.

Step 1: Compute the sine

Calculate the sine of -4.5 radians. In most calculators, ensure the mode is set to radians (not degrees), input -4.5 and press the sine function. Let this value be \( y \).

Step 2: Find the reciprocal

Calculate the reciprocal of \( y \), to find the cosecant (\( \csc \)). The reciprocal of any number \( n \) can be found by taking \( 1/n \). Similarly, the reciprocal of \( y \) is \( 1/y \). The result is the desired value for the cosecant of -4.5 radians, rounded to the nearest hundredth.

Key Concepts

Cosecant

The cosecant function is one of the six main trigonometric functions, which also include sine, cosine, tangent, secant, and cotangent. Specifically, the cosecant is the reciprocal of the sine function. While sine itself measures the ratio of the length of the opposite side to the hypotenuse in a right triangle, cosecant flips this ratio. Here's a simple way to remember:

• For an angle \( \theta \), \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

So, when we evaluate any angle's cosecant, we're looking for the value that when multiplied by the sine of that angle equals 1. Cosecant, like other trigonometric functions, is periodic, meaning it repeats its values over specific intervals.This can be particularly useful in various applications, such as signal processing or in studying wave patterns, where the properties of these functions are leveraged.

Radians

Radians are a unit of angular measure based on the radius of a circle. They are an alternative to degrees, which might be more familiar.The idea of radians is pretty straightforward:

• A full circle is \(2 \pi\) radians. This is roughly equivalent to 360 degrees.
• Thus, \( \pi \) radians is equal to 180 degrees.

When working with trigonometric functions in calculus or physics, radians are often preferred because they provide a more natural measure when dealing with certain mathematical models. Imagine that instead of counting the circle in a hundred degrees or in increments of ten, we measure in terms of how many radii fit around the circle's circumference. Many scientific calculators have a button to switch between degree and radian mode, which is crucial when calculating trigonometric values, as seen in exercises evaluating angles like \(-4.5\) radians.

Reciprocal Function

Reciprocal functions take any number and switch its positioning in a fraction setup, essentially turning it upside-down. When we refer to something as a reciprocal, we're hinting at a simple yet powerful operation. Here's how it works:

• If you have a number \( n\), its reciprocal is \( \frac{1}{n} \).

In trigonometry, this concept becomes useful when defining cosecant, secant, and cotangent, which are the reciprocals of sine, cosine, and tangent, respectively. Let's see an example:
If \( \sin(\theta) = y \), then \( \csc(\theta) = \frac{1}{y} \). The concept of reciprocals is fundamental in various areas of math because it neatly connects division to multiplication. Anytime you're dealing with reciprocal functions, you're leveraging this simple transformation to switch perspectives and solve different types of problems when basic operations give way to new approaches.