Problem 235 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \csc \frac{\pi}{4} $$

Step-by-Step Solution

Answer

The value of \( \csc \frac{\pi}{4} \) is \( \sqrt{2} \).

Step 1: Understand the Function

The function given is the cosecant, denoted as \( \csc \), which is the reciprocal of the sine function. That means \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Identify the Angle

We need to evaluate \( \csc \frac{\pi}{4} \). The angle \( \frac{\pi}{4} \) is 45 degrees in radians.

Step 3: Determine the Sine Value

Use the known trigonometric value \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 4: Calculate the Cosecant

Substitute the sine value into the cosecant definition: \( \csc \frac{\pi}{4} = \frac{1}{\sin \frac{\pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}} \).

Step 5: Simplify the Expression

Simplifying \( \frac{1}{\frac{\sqrt{2}}{2}} \) gives us \( 2 \times \frac{1}{\sqrt{2}} = \sqrt{2} \). Thus, \( \csc \frac{\pi}{4} = \sqrt{2} \).

Key Concepts

Cosecant

Cosecant is a trigonometric function that complements the sine function. It is often denoted as \( \csc \theta \) and is defined as the reciprocal of the sine. In simple terms, when you know the value of the sine of an angle, you can find the cosecant by taking the inverse of that value.

• Cosecant is defined as \( \csc \theta = \frac{1}{\sin \theta} \)
• It is an undefined value for angles where sine equals zero since division by zero is undefined.
• Such angles include multiples of \( \pi \) (180 degrees), where the sine function achieves a zero value.

Understanding how to compute the cosecant is crucial when dealing with trigonometric identities and solving equations involving this function.
Remember, the cosecant function is less commonly used compared to sine, cosine, and tangent, but it remains an important part of trigonometry.

Sine

The sine function is one of the most fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the opposite side over the hypotenuse.

• The formula is \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
• Sine values range between -1 and 1 for angles between 0 and 360 degrees, or \( 0 \) to \( 2\pi \) radians.
• For the angle \( \frac{\pi}{4} \) (or 45 degrees), the sine value is known to be \( \frac{\sqrt{2}}{2} \).

The function is periodic, meaning it repeats its values in regular intervals. This periodic nature is essential in many applications, including sound and light wave analysis.
Understanding the sine function allows you to handle various mathematical problems involving waveforms, heights, and rotations.

Radians

Radians are a unit of angle measurement that are essential in trigonometry. Unlike degrees, radians measure angles based on the radius of a circle.

• There are \( 2\pi \) radians in a full circle, equivalent to 360 degrees.
• The conversion formula is \( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \).
• The angle \( \frac{\pi}{4} \) radians is equal to 45 degrees as per this conversion.

Understanding radians can make it easier to work with the trigonometric functions in calculus and other advanced mathematics.
Using radians allows seamless integration with other mathematical concepts, particularly when dealing with periodic functions and oscillations.

Reciprocal Function

A reciprocal function is one where each input is transformed by taking its reciprocal (or inverse). This means that if you have a function \( f(x) = x \), its reciprocal would be \( g(x) = \frac{1}{x}\).

• Reciprocal functions are defined everywhere except where the input value is zero.
• For trigonometric functions, the reciprocal functions include cosecant, secant, and cotangent.
• The process for finding the reciprocal involves flipping the numerator and denominator.

In the context of trigonometry:

• The reciprocal of sine \( \sin \theta \) is cosecant \( \csc \theta = \frac{1}{\sin \theta} \).
• This relationship helps in solving trigonometric equations and proving identities.

Understanding reciprocal functions is key to mastering different trigonometric transformations and computations.