Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.
sin 38° 42'

Verified step by step guidance

1. First, convert the angle given in degrees and minutes to a decimal degree format. Recall that 1 minute is equal to \( \frac{1}{60} \) degrees. So, convert 42 minutes to degrees by calculating \( 42 \times \frac{1}{60} \).
2. Add the decimal degree value from the minutes to the whole degrees to get the total angle in decimal degrees: \( 38 + \text{(decimal from minutes)} \).
3. Use the sine function on your calculator with the angle in decimal degrees. Make sure your calculator is set to degree mode, not radians.
4. Calculate \( \sin(\text{decimal degrees}) \) using the calculator.
5. Round the result to six decimal places as requested.

Key Concepts

Angle Measurement in Degrees and Minutes

Angles can be expressed in degrees, minutes, and seconds, where 1 degree equals 60 minutes. To use a calculator, convert the angle from degrees and minutes to a decimal degree by dividing the minutes by 60 and adding to the degrees. For example, 38° 42' equals 38 + 42/60 = 38.7 degrees.

Using the Sine Function on a Calculator

The sine function relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. Calculators compute sine values using the angle in degrees or radians, so ensure the calculator is set to degree mode when inputting angles in degrees. This allows accurate evaluation of sin(38.7°).

Rounding and Precision in Calculations

When approximating trigonometric values, it is important to round the result to the specified number of decimal places for consistency and clarity. Here, answers should be rounded to six decimal places, which means keeping six digits after the decimal point, ensuring precision in the final answer.