Problem 39 - Calculus: One and Several Variables

Question

Evaluate to four decimal place accuracy. $$\sin 51^{\circ}$$.

Step-by-Step Solution

Answer

The sin of 51 degrees equals 0.7771 when rounded to four decimal places.

Step 1: Set Calculator to Degree Mode

Using mathematical software or a calculator, first set it to 'degree' mode. This is because the angle given in the problem is in degrees, and calculators often default to radians.

Step 2: Calculate Sine of 51 Degrees

Enter the number 51 into the calculator and press the 'sin' button. This gives a long decimal number.

Step 3: Round to Four Decimal Places

Round the resulting decimal number to four places. To do this, look at the fifth decimal place. If it's a number from 0 to 4, drop it and all numbers that follow. If it's a number from 5 to 9, add 1 to the fourth decimal place.

Key Concepts

Angle Measurement

The measurement of angles is a fundamental aspect of trigonometry. Angles can be measured in different units, with degrees and radians being the most common. In the context of basic trigonometry, degrees are often used because they are intuitive and straightforward. One complete rotation around a circle equals 360 degrees, making calculations more manageable for students initially learning these concepts.
Degrees are marked with the symbol \( ^\circ \), and it's crucial when solving problems involving trigonometric functions to know the unit of measurement. Incorrect units could lead to incorrect results, as trigonometric functions behave differently when angles are measured in degrees versus radians.

Degree Mode

When working with trigonometric functions like sine, cosine, or tangent, the mode of your calculator is significant. Most scientific calculators have two modes for angle measurement: degree and radian. Since angles can be provided in either of these units, it's essential to set the calculator to the correct mode to obtain accurate results.
To set your calculator to degree mode, look for a button or a setting that says 'MODE'. If your calculator isn't in degree mode by default, switch it by selecting 'DEG' or something similar. This ensures that when you input an angle like \(51^{\circ}\), it interprets it correctly, giving you the accurate sine value. This mode setting prevents common errors students might encounter when defaulting to radians.

Rounding Decimals

Rounding decimals is often required to present a more readable answer or when the problem specifies a certain precision. When rounding to four decimal places, only the first four digits after the decimal point are considered. The process involves looking at the fifth digit.
- If the fifth digit is between 0 and 4, you leave the fourth digit unchanged. - If the fifth digit is between 5 and 9, increase the fourth digit by one.
For example, if your sine function calculation results in the value 0.7778513, you would look at the fifth decimal place, which is 5. Since 5 is greater than or equal to 5, you increase the fourth decimal place from 8 to 9, giving you a rounded result of 0.7779. Rounding ensures consistency, especially in academic settings where precise answers are required.

Trigonometry

Trigonometry is a branch of mathematics dealing with the relationships between the angles and sides of triangles. The sine function, abbreviated as 'sin', is one of the six fundamental trigonometric functions used primarily to relate an angle in a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse.
These relationships can extend beyond right triangles and are pivotal in understanding oscillatory behavior, waves, and various real-world applications. Understanding that the sine of 51 degrees is approximately 0.7779 after rounding helps visualize how a particular angle translates into a ratio.
Trigonometry is not just about angles and ratios. It's about understanding functions that model periodic phenomena, which is essential in fields like physics, engineering, and computer science. It lays the groundwork for more advanced mathematical studies.