Problem 15 - Basic Engineering Mathematics

Question

Problem 15. Evaluate correct to 4 decimal places: (a) sine \(11^{\circ}\) (b) sine \(121.68^{\circ}\) (c) sine \(259^{\circ} 10^{\prime}\) (a) sine \(11^{\circ}=\mathbf{0 . 1 9 0 8}\) (b) sine \(121.68^{\circ}=\mathbf{0 . 8 5 1 0}\) (c) sine \(259^{\circ} 10^{\prime}=\operatorname{sine} 259 \frac{10^{\circ}}{60}=-\mathbf{0 . 9 8 2 2}\)

Step-by-Step Solution

Answer

(a) 0.1908, (b) 0.8510, (c) -0.9822.

Step 1: Convert Degrees to Radians

To evaluate the sine of an angle, first convert the degree measurement into radians. The formula to convert degrees to radians is: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This step is crucial for calculating trigonometric functions on a calculator without a degree mode.

Step 2: Use Calculator for Sine Function

After converting the angle to radians, use a scientific calculator to find the sine of the angle. Ensure your calculator is in the radian mode. Input the radian value and press the sine function key to get the result.

Step 3: Round the Answer

After obtaining the sine value from the calculator, round off the result to four decimal places for precision. This rounding step ensures the answer meets the requirements of the exercise.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions commonly used in mathematics. It is symbolized as "sin" and is particularly used to relate the angles of a right triangle to the ratios of its sides. In a right triangle:

• The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

For a given angle θ, this can be expressed as: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]The sine function is periodic, meaning it repeats its values in regular intervals every 360 degrees or 2π radians. It takes input values from real numbers (which could represent angles in degrees or radians) and outputs values ranging from -1 to 1.

This function is extensively used in various applications like physics, engineering, and even in music theory because it models wave patterns naturally.

Degree to Radian Conversion

Degrees and radians are units used to measure angles, and each has its particular uses depending on the context. Degree to radian conversion becomes essential when dealing with trigonometric calculations on calculators that operate in radian mode rather than degree mode. To convert degrees to radians, use the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]Pi (\(\pi\)) here is approximately 3.14159, representing the ratio of the circumference of a circle to its diameter. Thus, 180 degrees is equivalent to \(\pi\) radians.

For example:

• To convert 11 degrees into radians, calculate: 11 × \(\frac{\pi}{180} \approx 0.191 \) radians.
• For 121.68 degrees, it converts to: 121.68 × \(\frac{\pi}{180} \approx 2.124 \) radians.
• For angles given in degrees and minutes, like 259° 10', first convert the minutes to a decimal degree: 259.1667, then multiply by \(\frac{\pi}{180}\).

Understanding this conversion ensures accuracy in computations and allows the use of trigonometric functions without switching calculator modes.

Scientific Calculator Usage

A scientific calculator is a powerful tool that helps perform complex mathematical calculations efficiently and accurately. When calculating the sine of an angle, ensure the calculator is in the correct mode (radian or degree) based on your angle measurement. To use a scientific calculator for the sine function:

• First, determine the mode needed for the angle. For radian measures, the calculator must be in radian mode. For degree measures, it should be in degree mode.
• Once your calculator is set to the correct mode, input the angle value.
• Press the "sin" button to get the sine of the angle displayed.

Let’s say you're working with an angle converted to radians; simply input it and use the "sin" function. Ensure to check whether the calculator's display mode aligns with what's required for the solution. After obtaining the sine value, if required, round it to four decimal places to meet the precision specified in problems like textbook exercises. This ensures clarity and consistency with the expected results.