Problem 38 - Algebra 2

Question

Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. $$ \csc 130^{\circ} $$

Step-by-Step Solution

Answer

After converting \( 130^{\circ} \) into radians and finding the sine of that value, taking the reciprocal of this sine value will give the \( \csc \) of \( 130^{\circ} \).

Step 1: Convert the Degrees to Radians

Trigonometric functions in calculus typically operate on angles measured in radians, so convert \( 130^{\circ} \) to radians. To do this, multiply \( 130^{\circ} \) by \( \frac{\pi}{180} \).

Step 2: Find the Sine Value

Now determine the sine of the radian value obtained in step 1. The sine function on a unit circle is defined as the length of the vertical line from the origin of the circle to a point on its circumference. It can be determined using various trigonometric tables or a calculator.

Step 3: Find the Cosecant (Reciprocal of Sine)

The csc function, or cosecant, is the reciprocal of the sine function. So, to get the cosecant value, we'll take the reciprocal of the sine value found in step 2. If the sine value is represented as \( m \), then the cosecant (\( \csc \)) will be \( \frac{1}{m} \).

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are special types of functions that are closely related to the basic trigonometric functions: sine, cosine, and tangent. The main reciprocal functions are cosecant (csc), secant (sec), and cotangent (cot). They are important because they can help solve a variety of mathematical problems involving angles and lengths in right triangles and circles.

Here's how these functions relate to their primary counterparts:

• Cosecant ( \( \csc \theta \) ): It is the reciprocal of the sine function. In formula terms, \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant ( \( \sec \theta \) ): It is the reciprocal of the cosine function. Therefore, \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent ( \( \cot \theta \) ): It is the reciprocal of the tangent function. This is given by \( \cot \theta = \frac{1}{\tan \theta} \).

Remember, since these functions are reciprocals, they will be undefined wherever their primary counterparts are zero. Thus, understanding when and why these values become undefined is key when using reciprocal trigonometric functions.

Radian Conversion

The conversion between degrees and radians is a fundamental concept in trigonometry because many advanced mathematics and calculus problems use radians instead of degrees. It is important for students to become comfortable converting between these units.

To convert from degrees to radians, multiply the degree value by \( \frac{\pi}{180} \). This is because there are \( 180 \) degrees in\( \pi \) radians.

• For example, to convert \( 130^{\circ} \) to radians, you calculate: \[ 130^{\circ} \times \frac{\pi}{180} = \frac{130 \pi}{180} = \frac{13 \pi}{18} \].

This conversion is essential when working with trigonometric functions on a unit circle, as most trigonometric calculators and many trigonometric tables use radians.

Unit Circle

The unit circle is a powerful tool in trigonometry. It is a circle with a radius of 1 centered at the origin of a coordinate plane. It helps in understanding the properties of trigonometric functions at various angles.

On the unit circle, each angle corresponds to a point \((x, y)\) on the circle. Here:

• The x-coordinate represents \( \cos \theta \)
• The y-coordinate represents \( \sin \theta \)

Because the radius is 1, the coordinates of any point on the unit circle directly give the values of cosine and sine for any angle (in radians).

This simple yet effective method makes it easy to find regular and reciprocal trigonometric values, such as cosecant, as it is simply the reciprocal of the y-coordinate, or \( \frac{1}{\sin \theta} \).

Therefore, understanding how to use the unit circle efficiently can drastically simplify solving trigonometry problems.

Trigonometric Tables

Trigonometric tables were essential tools before the advent of calculators. They provided values for trigonometric functions for different angles, usually in increments of one degree. Though less common now due to digital tools, knowing how to use them can be useful.

In these tables:

• Sine and cosine values are directly listed for each angle.
• Reciprocal functions like cosecant can be determined by finding the sine value and simply taking the reciprocal.

For angles that fall between those listed in the table, interpolation might be needed. This requires estimating a value between two known values.

Understanding trigonometric tables helps appreciate the history of mathematics and enhances numerical literacy, providing insight into how angles relate to the sides of right triangles and circular measurements.