Problem 23 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\csc 45^{\circ}$$

Step-by-Step Solution

Answer

The exact value is \( \sqrt{2} \), approximately 1.414.

Step 1: Recall the Definition of Cosecant

The cosecant of an angle \( \theta \) is defined as the reciprocal of the sine of that angle. Therefore, \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Find \( \sin 45^{\circ} \)

The sine of \( 45^{\circ} \) is a well-known value: \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

Step 3: Calculate \( \csc 45^{\circ} \)

Substitute the value found in Step 2 into the definition of cosecant: \( \csc 45^{\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} \). Simplify this expression to obtain the exact value: \( \csc 45^{\circ} = \sqrt{2} \).

Step 4: Use Calculator for Decimal Approximation

The exact value \( \sqrt{2} \) is irrational. Use a calculator to find the decimal approximation of \( \sqrt{2} \). This gives approximately 1.41421356.

Key Concepts

Cosecant

To really grasp the cosecant function, it's helpful to think about its relationship with the sine function. Trigonometric functions often appear in pairs that are reciprocals, and the cosecant (abbreviated as csc) is no exception. It is, in fact, the reciprocal of the sine function. Here's the formula to remember:

• For any angle \( \theta \), the cosecant is \( \csc \theta = \frac{1}{\sin \theta} \).

It's important to note that cosecant is undefined for angles where the sine value is zero. For example, \( \csc 0^{\circ} \) or \( \csc 180^{\circ} \) don't exist because you cannot divide by zero (since \( \sin 0^{\circ} = 0 \)).
This function can be particularly useful in various calculations, especially in trigonometry problems involving right triangles, waves, or oscillations. Understanding that the cosecant is just about flipping the sine value can make it relatable and easier to use whenever you need it.

Sine of 45 degrees

The sine of 45 degrees is a fundamental trigonometric value that pops up frequently due to its simplicity and the symmetry it represents in a 45-degree right triangle. This angle is a part of the special 45-45-90 triangle, where both non-right angles are equal.
The sine of 45 degrees can be expressed as:

• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)

This expression is derived from considering an isosceles right triangle, where both legs are equal in length, and utilizing the Pythagorean theorem. The sine function, being the ratio of the opposite side to the hypotenuse, yields the value of \( \frac{\sqrt{2}}{2} \).
Understanding \( \sin 45^{\circ} \) is essential as it often appears in standardized tests, engineering, physics, and any math problems involving angles or rotations. It simplifies many calculations since it bridges the relationship between angles and sides in right triangles.

Decimal Approximation

Decimal approximation is crucial in mathematics when dealing with irrational numbers like \( \sqrt{2} \). While the exact value of \( \sqrt{2} \) is very precise and exact, it is often necessary to convert this into a more practical form for computational purposes.
When you need a decimal approximation for \( \sqrt{2} \), here's what you get:

• \( \sqrt{2} \approx 1.41421356 \)

This decimal form makes it easier to work with in calculations, especially when you don’t have access to exact arithmetic tools.
In real-life applications, such as engineering and physics, you'll often need to round this approximation. However, remember that rounding can introduce small errors, so it's important to retain as many decimal places as are necessary for precision in your work. Calculators and software can make this process seamless, ensuring you have both numerical efficiency and accuracy.