Problem 42 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the trigonometric functions. Make sure the calculator is in DEGREE mode. Round to four decimal places. $$\csc 36^{\circ}$$

Step-by-Step Solution

Answer

The rounded result of \(\csc(36^{\circ})\) is your output from Step 3. Note that this instruction does not warrant a predefined answer as the output might vary slightly due to the variance in calculator round-offs

Step 1: Switch the calculator to DEGREE mode

To begin, ensure the calculator is in DEGREE mode. This is typically achieved by finding a button labeled 'mode' or 'setup' and then selecting 'DEGREE'. It's crucial that the calculator is in this mode since we are working with degrees, not radians.

Step 2: Calculate the Sine of 36 degrees

Next, input 36 into the calculator, then press the 'sin' button (or vice-versa depending on your calculator model). This will output the sine of 36 degrees.

Step 3: Calculating the Cosecant

Since the cosecant is the reciprocal of the sine, find the reciprocal of the result from Step 2. You can do this by typing '1 /', followed by the result. Round the result to four decimal places as directed.

Key Concepts

Cosecant (csc)

The cosecant function, often abbreviated as 'csc,' is one of the six primary trigonometric functions. It is defined as the reciprocal of the sine function, meaning that for any angle \theta, the cosecant is given by:
\[\csc(\theta) = \frac{1}{\sin(\theta)}\]
This relationship implies that anytime you have the sine of an angle, you can find the cosecant by simply taking the inverse of the sine value. In the context of a right triangle, the cosecant of an angle is the length of the hypotenuse divided by the length of the opposite side. However, it's worth noting that the cosecant isn't as commonly used in calculations as sine, hence it's typically not directly accessible on basic calculators and requires a bit more effort to compute.

Sine Function

The sine function is fundamental in trigonometry and represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. For any given angle \theta in a right-angled triangle:
\[\sin(\theta) = \frac{\text{Opposite side}}{\text{Hypotenuse}}\]
When you input an angle into a scientific calculator and press the 'sin' function, it calculates this ratio for you. In the context of our example with 36 degrees, the calculator uses the sine function to determine the ratio associated with a 36-degree angle in a unit circle or a right-angle triangle. The sine of an angle is dimensionless since it's a ratio, and it always falls between -1 and 1.

Degree Mode

Scientific calculators often have the capacity to operate in various modes, with 'degree' and 'radian' modes being the most pertinent for trigonometry. Degree mode is used when you are working with angles measured in degrees. One complete revolution around a circle is 360 degrees, which is equivalent to 2π radians. It's crucial for accuracy to set the calculator to the correct mode corresponding to the type of angle measurement you're using.

Activating Degree Mode

To switch to degree mode, look for a 'mode' or 'setup' button on your scientific calculator. Navigate through the options until you find the setting for angle measurements, then select 'degree' to ensure that the trigonometry functions interpret the angles you enter as degrees.

Scientific Calculator

A scientific calculator is an advanced calculator that's capable of performing complex mathematical functions beyond addition, subtraction, multiplication, and division. It's an essential tool for students and professionals in science, engineering, and mathematics. Here are some typical features:

• Trigonometric functions such as sine, cosine, and tangent, along with their inverses
• Exponential and logarithmic functions
• Statistical functions
• Ability to switch between degree and radian modes for angle measurements

Using a Scientific Calculator for Trigonometry

For trigonometric calculations, make sure that the calculator is in the correct mode (degree or radian). Input the angle, press the corresponding trigonometric function key, and the calculator will display the result. To calculate functions like 'cosecant' that are not directly available, you have to use the appropriate reciprocal or inverse function of the available keys.