Problem 49 - Precalculus: A Concise Course

Question

Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct angle mode.) (a) \(\sin 16.35^{\circ}\) (b) \(\csc 16.35^{\circ}\)

Step-by-Step Solution

Answer

The evaluation results rounded to four decimal places are: (a) \(\sin 16.35^{\circ} = 0.2826\) (b) \(\csc 16.35^{\circ} = 3.5405\)

Step 1: Set Calculator to Degree Mode

Before starting the calculation, make sure the calculator is set to 'Degree' mode. Usually, this can be done in the settings of the calculator. This is essential because the given angles are in degrees.

Step 2: Calculate \(\sin 16.35^{\circ}\)

After setting the calculator to the right mode, input '\(\sin 16.35\)'. The calculator will return a value, which then should be rounded to four decimal places.

Step 3: Calculate \(\csc 16.35^{\circ}\)

Cosecant is the reciprocal of the sine function. Therefore, after computing '\(\sin 16.35^{\circ}\)', take the reciprocal of this value to find '\(\csc 16.35^{\circ}\)'. Again, round the result to four decimal places.

Key Concepts

Understanding the Sine Function

The sine function is one of the basic trigonometric functions often encountered in mathematics. It relates an angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. Given an angle \( \theta \), the sine function can be expressed as:

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)

When using the sine function in a calculator, remember you are finding this ratio given the angle. For specific angles, the value is usually a fixed decimal, like \( \sin 30^{\circ}= 0.5 \). Therefore, whenever asked to calculate the sine of a particular angle, like \( 16.35^{\circ} \), you follow similar principles.
This function is periodic, with a cycle repeating every \(360^{\circ}\), meaning it’s particularly helpful in periodic scenarios, such as waves and oscillations.

Exploring the Cosecant Function

The cosecant function is another important trigonometric function. It is the reciprocal of the sine function. Hence, it is defined as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This implies that whenever you calculate the sine of an angle, you can easily find the cosecant by taking the reciprocal of that sine value. For example, if you know \( \sin 16.35^{\circ} \) is some value \( x \), then \( \csc 16.35^{\circ} = \frac{1}{x} \).
Bear in mind, the cosecant function grows positively larger and shrinks negatively smaller as the sine approaches zero, except it remains undefined when the sine is zero because you cannot divide by zero. This characteristic makes it valuable in more complex trigonometric analysis.

Navigating Degree Mode in Calculators

Degree mode is a crucial setting on scientific calculators, especially when dealing with trigonometry. Basically, it dictates how the calculator interprets the angles you’re entering. There are generally two modes:

• Degree mode (°)
• Radian mode (rad)

In most high school or early college math problems, angles are usually expressed in degrees, similar to how a circle is 360 degrees.
Therefore, before solving problems like \( \sin 16.35^{\circ} \) or \( \csc 16.35^{\circ} \), ensure the calculator's mode is set to "Degree". This adjustment is typically found in the calculator's mode settings or function button. Stepping through this correctly ensures your results are accurate for problems expressed in degrees.

Utilizing Calculators for Trigonometric Functions

Calculators are excellect tools for quickly evaluating trigonometric functions like sine and cosecant. Follow these steps for optimal results:

• Ensure the calculator is in the correct angle mode (Degree or Radian).
• Input the angle value; for example, \( \sin 16.35 \).
• Read the output and round it accordingly when instructed, such as to four decimal places.
• For cosecant, take your result from the sine and compute \( \frac{1}{\text{value}} \).

Always double-check that every step is followed, as missteps can lead to wrong answers. Calculators can handle these computations precisely when used correctly, ensuring your results are both quick and accurate.