Problem 33 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \csc 125^{\circ} $$

Step-by-Step Solution

Answer

\( \csc 125^{\circ} \approx 1.2209 \).

Step 1: Understanding the Function

The question is asking for the cosecant of an angle, which is the reciprocal of the sine function. Specifically, \ \( \csc \theta = \frac{1}{\sin \theta} \) where \( \theta \) is the angle given in degrees.

Step 2: Convert Degrees to Radians

Although it's not necessary in this case, it's often useful to know that to convert degrees to radians, use the formula \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \). For \( 125^\circ \), this conversion isn't needed as typical calculators can work directly with degrees.

Step 3: Find the Sine Value

Calculate the sine of \( 125^\circ \) using a calculator. You should get: \ \( \sin 125^\circ \approx 0.8192 \) (rounded to four decimal places).

Step 4: Calculate the Cosecant

Using the reciprocal relationship, calculate cosecant by taking one over the sine value: \ \( \csc 125^\circ = \frac{1}{\sin 125^\circ} = \frac{1}{0.8192} \approx 1.2209 \), rounded to four decimal places.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are fundamental in the study of trigonometry and are widely used in various fields such as physics, engineering, and architecture. The main trigonometric functions include:

• Sine (\( ext{sin} \)), which relates the opposite side to the hypotenuse of a right-angled triangle.
• Cosine (\( ext{cos} \)), relating the adjacent side to the hypotenuse.
• Tangent (\( ext{tan} \)), the ratio of the opposite to the adjacent side.

There are also three additional functions known as the reciprocal trigonometric functions, which are the focus of some specific problems, like the exercise above with the cosecant function.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the primary trigonometric functions and are useful for various calculations in trigonometry. Each basic function has its reciprocal counterpart:

• Cosecant (\( ext{csc} \)) is the reciprocal of sine: \[ ext{csc} \, \theta = \frac{1}{\text{sin} \, \theta} \]
• Secant (\( ext{sec} \)) is the reciprocal of cosine: \[ ext{sec} \, \theta = \frac{1}{\text{cos} \, \theta} \]
• Cotangent (\( ext{cot} \)) is the reciprocal of tangent: \[ ext{cot} \, \theta = \frac{1}{\text{tan} \, \theta} \]

These functions are particularly useful in solving complex trigonometric problems where inverses of values are needed. In the original exercise, we found \( \text{csc} \, 125^\circ \) using this relationship. By first calculating \( \text{sin} \, 125^\circ \), we can easily determine its reciprocal, the cosecant.

Angle Measurement in Degrees

Angles can be measured in different units, typically degrees or radians. Degrees are much more familiar in everyday applications. One full revolution around a circle is 360 degrees. However, in mathematics, radians are often employed for more advanced calculations. The conversion between degrees to radians is given by the formula:\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]For example, to convert 125 degrees to radians, you multiply:\[125 \times \frac{\pi}{180} \approx 2.1817 \, \text{radians}\]Even though the conversion wasn't necessary in the original exercise because calculators can handle degrees directly, this understanding is critical. Degrees offer a direct and intuitive way to measure angles, which is why they are often used in many practical situations, like navigation and layout design, in addition to trigonometric applications.