Problem 22 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \sin 57^{\circ} 40^{\prime} $$

Step-by-Step Solution

Answer

The sine of \(57^{\circ} 40^{\prime}\) is approximately 0.8413.

Step 1: Understand the Problem

We need to find the sine of the angle given in degrees and minutes: \(57^{\circ} 40^{\prime}\). This involves converting the angle to a format that can be used with the sine function.

Step 2: Convert Minutes to Decimal Degrees

Since the angle is given in degrees and minutes, convert 40 minutes into decimal degrees using the formula: \(\text{minutes}\times\frac{1}{60}\). \[ 40^{\prime} \times \frac{1}{60} = 0.6667^{\circ} \]

Step 3: Add to Degrees

Add the converted minutes to the degrees:\[ 57^{\circ} + 0.6667^{\circ} = 57.6667^{\circ} \]

Step 4: Calculate the Sine Value

Use a calculator to find the sine of the angle \(57.6667^{\circ}\). Make sure your calculator is set to degrees. \(\sin 57.6667^{\circ} \approx 0.8413\)

Step 5: Conclusion

The sine of \(57^{\circ} 40^{\prime}\) approximates to four decimal places as 0.8413.

Key Concepts

Degree-Minute Conversion

When dealing with angles, sometimes they're expressed in both degrees and minutes. This can be a bit confusing, but it's quite simple to understand.
For angles expressed this way, there are 60 minutes in one degree. So, if you have an angle like \(57^{\circ} 40^{\prime}\), the 40 minutes need to be converted into decimal degrees to make calculations easier.
To do this conversion, use the formula:

• Multiply the number of minutes by \(\frac{1}{60}\).

In our case:

• \(40^{\prime} \times \frac{1}{60} = 0.6667^{\circ}\).

This means that \(57^{\circ} 40^{\prime}\) is the same as \(57.6667^{\circ}\) when using decimal degrees which is essential for trigonometric calculations.

Sine Function

The sine function is a fundamental part of trigonometry, often used to relate angles with their ratios in right triangles.
The sine of an angle \(\theta\) in a right-angled triangle is defined as the ratio of the length of the side opposite to the angle, to the length of the hypotenuse.
Mathematically, this is expressed as:

• \(\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\)

For angles greater than \(0^{\circ}\) and up to \(90^{\circ}\) (also known as acute angles), the sine function gradually increases from 0 to 1.
When using a calculator for sine values, ensure it's set to degrees if your angle is in degrees, as this affects accuracy.

Decimal Degree Calculation

Decimal degrees offer a simplified approach to working with angles in trigonometry, particularly when using calculators and other mathematical tools.
Converting an angle into decimal degrees involves breaking it down from its traditional "degrees and minutes" form into a single numerical representation.
Take the previous example of \(57^{\circ} 40^{\prime}\). After conversion:

• The 40 minutes become 0.6667 degrees when calculated using \(\text{minutes} \times \frac{1}{60}\).

Thus, the angle becomes \(57.6667^{\circ}\).Once in this form, these decimal degrees can be directly inputted into a calculator for finding trigonometric functions like sine, cosine, and tangent with greater ease and accuracy.
This simplification aids in reducing potential errors during calculations.