Problem 19 - Precalculus: Functions and Graphs

Question

Approximate to three decimal places. (a) \(\sin 73^{\circ} 20^{\prime}\) (b) \(\cos 0.68\)

Step-by-Step Solution

Answer

(a) \( \sin 73^{\circ} 20' \approx 0.955 \); (b) \( \cos 0.68 \approx 0.777 \).

Step 1: Understanding the Problem

We need to find approximations for the trigonometric values \(\sin 73^{\circ} 20'\) and \(\cos 0.68\) to three decimal places. A calculator or trigonometric tables may be used to find accurate values.

Step 2: Converting Degrees and Minutes to Decimal Degrees for (a)

The angle for part (a) is given in degrees and minutes: \(73^{\circ} 20'\). To convert this to decimal degrees: Convert the minutes to degrees by dividing by 60: \(20' = \frac{20}{60} = 0.3333\ldots^{\circ}\) Add this to the degrees: \(73^{\circ} + 0.3333^{\circ} = 73.3333^{\circ}\) So, \(73^{\circ} 20' = 73.3333^{\circ} \) in decimal form.

Step 3: Calculating \(\sin 73.3333^{\circ}\)

Using a calculator set to degree mode, calculate \(\sin 73.3333^{\circ}\): \(\sin 73.3333^{\circ} \approx 0.955\) This value is rounded to three decimal places.

Step 4: Calculation for \(\cos 0.68\)

The angle in part (b) is given in radians (0.68). Using a calculator set to radian mode, calculate \(\cos 0.68\): \(\cos 0.68 \approx 0.777\) Again, this value is rounded to three decimal places.

Key Concepts

Sin Function

The sine function, represented as \( \sin \), is a fundamental concept in trigonometry. It deals with the relationship between the angles and lengths of a right-angled triangle. The sine of an angle is a ratio. It is the length of the opposite side to that angle divided by the length of the hypotenuse.
For any angle \( \theta \), you can calculate:

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

Learning how to apply this function is crucial for solving problems involving triangle measurements. The sine function is periodic, and its values range from -1 to 1. It is frequently used in oscillations, waves, and circle-related problems.
When you use a calculator to find \( \sin 73^{\circ} 20' \), remember to convert the angle to a decimal degree first. This ensures accuracy in results.

Cos Function

The cosine function, symbolized as \( \cos \), is another essential trigonometric function. Similar to sine, it relates the angles and sides of a right triangle. However, cosine uses the adjacent side and the hypotenuse.
The formula is as follows:

• \( \cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)

The cosine function gives values from -1 to 1, depending on the angle. It is invaluable in calculating distances, speeds, and is widely used in physics and engineering.
For an angle like \( 0.68 \) radians, you must ensure your calculator is set to the correct mode (radian mode). This allows you to get an accurate decimal approximation.

Degree to Radian Conversion

Understanding how to convert between degrees and radians is fundamental in trigonometry. Degrees and radians are two units that measure angles. Often, problems in math or science may require conversion between these units.
The relationship is defined by the fact that \( 180^{\circ} \) equals \( \pi \) radians. Therefore, the conversion formulas are:

• From degrees to radians: \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \)
• From radians to degrees: \( \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} \)

Converting is straightforward when knowing these formulas. Using them ensures that angle measurements are suitable for trigonometric calculations.

Decimal Approximation

Decimal approximation is used to express numbers more simply while maintaining accuracy within acceptable limits. This process is common in mathematics, especially when dealing with irrational numbers or when exact values are impractical.
To approximate a number to a specific number of decimal places means to round it to that precision. For example:

• \( \sin 73.3333^{\circ} \approx 0.955 \)
• \( \cos 0.68 \approx 0.777 \)

These have been rounded to three decimal places, making them easier to work with. This rounding helps ensure that calculations are manageable and provides clarity without losing significant accuracy.