Problem 3 - Technical Mathematics with Calculus

Question

Evaluate each trigonometric expression to three significant digits. $$3.72\left(\sin 28.3^{\circ}+\cos 72.3^{\circ}\right)$$

Step-by-Step Solution

Answer

Evaluate \(\sin 28.3^\circ \approx 0.474\), \(\cos 72.3^\circ \approx 0.303\), sum to get \(0.474 + 0.303 = 0.777\), and multiply by 3.72 to find \(3.72 \times 0.777 \approx 2.89\).

Step 1: Evaluate the Sine Function

First, find the value of the sine function for the given angle. Evaluate \(\sin 28.3^\circ\) using a calculator or a trigonometric table.

Step 2: Evaluate the Cosine Function

Next, find the value of the cosine function for the given angle. Evaluate \(\cos 72.3^\circ\) in a similar manner, using a calculator or a trigonometric table.

Step 3: Sum the Function Values

Sum the results of \(\sin 28.3^\circ\) and \(\cos 72.3^\circ\) to find the total value inside the parentheses.

Step 4: Multiply by the Given Constant

Multiply the sum from Step 3 by the constant 3.72 to get the final answer.

Step 5: Round to Three Significant Digits

Round the result from Step 4 to three significant digits to adhere to the precision requirement of the exercise.

Key Concepts

Sine Function

The sine function is one of the basic functions in trigonometry. Representing the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle, the sine function is crucial for solving problems involving right-angled triangles. When you encounter an expression like \(\sin 28.3^\circ\), it's asking for the sine of the angle measured in degrees.

In practical terms, to evaluate \(\sin 28.3^\circ\), you would use a scientific calculator or a trigonometric table. Ensure your calculator is set to degrees if you're using one. For an accurate trigonometric computation up to three significant digits, it’s paramount to input the angle correctly and understand how your calculator displays results.

Cosine Function

The cosine function, commonly denoted as \(\cos\), is another fundamental trigonometric function. It represents the ratio of the length of the adjacent side to the hypotenuse of a right-angled triangle. When you're asked to find \(\cos 72.3^\circ\), you're looking for the cosine of the angle in degrees.

Like the sine function, the cosine value for a specific angle can be found using a scientific calculator or looked up in trigonometry tables. Accurate use of these tools is essential. Ensure that you are also in the correct mode for degree measurement when inputting the value into your calculator to obtain a precise result.

Significant Digits

Significant digits, also known as significant figures, are a way of expressing precision in numerical answers. These digits include all non-zero numbers, zeros between non-zero numbers, and any zero at the end of a decimal. Understanding significant digits is crucial in ensuring your answers reflect the correct level of precision as required by a problem.

In the context of our trigonometric expression, after computing the value of \(3.72(\sin 28.3^\circ + \cos 72.3^\circ)\), the next step is to round this answer to three significant digits. This means you'll keep the first three digits that carry meaning given their position and magnitude, and adjust the final one by rounding up or down based on the value that follows. The rounded value effectively communicates the precision of the computed result.