Problem 38 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \csc 347^{\circ} $$

Step-by-Step Solution

Answer

\( \csc 347^{\circ} \approx -6.3925 \)

Step 1: Understand the function

The problem requires us to find the value of the cosecant function, denoted as \( \csc \). The cosecant function is the reciprocal of the sine function, meaning \( \csc x = \frac{1}{\sin x} \).

Step 2: Convert degrees to radians

Although we often work with degrees, some calculators and mathematical operations use radians. Let's calculate \( \csc 347^{\circ} \) by first ensuring that \( 347^{\circ} \) is in degrees. No conversion to radians is necessary for this problem since we will input the degree value directly into a calculator.

Step 3: Calculate the sine of the angle

We need to compute \( \sin 347^{\circ} \). Use a calculator set to degree mode to find this value. \( \sin 347^{\circ} \approx -0.1564 \).

Step 4: Calculate the cosecant of the angle

Now that we know \( \sin 347^{\circ} \approx -0.1564 \), calculate the cosecant by taking the reciprocal: \( \csc 347^{\circ} = \frac{1}{\sin 347^{\circ}} \).

Step 5: Perform the reciprocal calculation

Take the reciprocal of \(-0.1564\) to get \( \csc 347^{\circ} \approx -6.3925 \).

Step 6: Verify the result

Double-check the reciprocal calculation. Calculate \( \frac{1}{-0.1564} \) to ensure the accuracy of \( \csc 347^{\circ} \approx -6.3925 \).

Key Concepts

Cosecant Function

The cosecant function is one of the important trigonometric functions and is denoted as \( \csc \). It is defined as the reciprocal of the sine function. In mathematical terms, this means:

• \( \csc x = \frac{1}{\sin x} \)

If you know the value of \( \sin x \), you can easily find \( \csc x \) by taking its reciprocal. It is crucial to be cautious about the sign and value of \( \sin x \) because the cosecant function becomes undefined whenever \( \sin x = 0 \). This is because division by zero is not possible in mathematics.
When using the cosecant function, you must ensure your calculator is set to the correct mode (degree or radian) as required by the problem. In our example of \( \csc 347^{\circ} \), the degree mode is appropriate, making it straightforward to first find \( \sin 347^{\circ} \) before calculating its reciprocal.

Reciprocal Functions

Reciprocal functions flip a number, converting it from \( a \) to \( \frac{1}{a} \). In trigonometry, several functions have reciprocal counterparts:

• \( \sin \rightarrow \csc \)
• \( \cos \rightarrow \sec \)
• \( \tan \rightarrow \cot \)

When using reciprocal functions, it’s essential to note that they can become undefined if the original function equals zero. For example, \( \csc x \) becomes undefined when \( \sin x = 0 \), such as at multiples of \( 180^{\circ} \) (e.g., \( \sin 0^{\circ} = 0 \), hence \( \csc 0^{\circ} \) is undefined).

Reciprocals are a valuable tool in solving certain types of problems where direct function values are easier to work with. Understanding this concept can simplify and streamline trigonometric calculations.

Angle Measurement in Degrees

Angles can be measured in degrees or radians, but most introductory problems and everyday contexts use degrees. In degree measurement:

• A full circle is \( 360^{\circ} \).
• A right angle is \( 90^{\circ} \).
• Angles beyond \( 360^{\circ} \) repeat the cycle of a circle.

When performing trigonometric calculations, either mode can be used, but it must correspond with the problem’s requirements. For the exercise of \( \csc 347^{\circ} \), the angle is already given in degrees. Ensure your calculator is in degree mode to avoid errors.
Understanding how angles work, especially ones larger than \( 360^{\circ} \), is crucial. When you see an angle like \( 347^{\circ} \), it lies in the fourth quadrant of the unit circle where sine values are negative. This fact connects beautifully to the operations performed during calculating \( \csc 347^{\circ} \). Knowing these fundamentals makes handling trigonometric functions more intuitive.