Problem 43 - Calculus: One and Several Variables

Question

Evaluate to four decimal place accuracy. $$\tan 72.4^{\circ}$$.

Step-by-Step Solution

Answer

The value of \( \tan 72.4^{\circ}\) to four decimal places is 3.0777.

Step 1: Convert Degree to Radians

Before the calculation can be carried out, the angle must first be converted from degrees to radians, as scientific calculators use radians for trigonometric functions. The conversion formula is \(Radians = Degree * (\pi /180)\). Therefore, \(72.4^{\circ} = 72.4 * (\pi /180) = 1.26321\) radians (when rounded up to 5 decimal places).

Step 2: Calculate Tangent

Next, apply the tangent function to the converted angle. Use a calculator or scientific computing software to find \( \tan(1.26321)\).

Step 3: Round to Four Decimal Places

Finally, take the result from the previous step and round it off to four decimal places. This gives the final result.

Key Concepts

Degree to Radian Conversion

When you work with trigonometric functions like sine, cosine, and tangent, converting degrees to radians can be very important. Calculators typically assume angles are in radians unless instructed otherwise. To convert from degrees to radians, you use the formula:\[\text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right)\]This formula utilizes the fact that \(\pi\) radians is equal to 180 degrees, which is equivalent to half a circle. Let's put this into context with our example where we convert 72.4 degrees to radians:- Multiply 72.4 by \(\frac{\pi}{180}\)- This gives \(1.26321\) radians (to five decimal places). This conversion allows us to use the trigonometric functions as they are intended in calculators.

Tangent Function

The tangent function, designated as \(\tan(\theta)\), relates to the angle \(\theta\) in a right-angled triangle. It defines the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.For an angle \(\theta\) measured in radians:- \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)In more advanced mathematics, the tangent function can be extended beyond triangles to all real numbers via the unit circle. It is periodic with a period of \(\pi\).Using the earlier calculated radian measure of 1.26321, you plug this value into a calculator to find \(\tan(1.26321)\). This calculated value gives you a precise representation of the angle's tangent in trigonometric calculations.

Rounding Numbers

Rounding numbers is an essential mathematical skill, often required to ensure results are both meaningful and manageable. This is especially true in fields where precision is essential, yet practicality requires concise expression. To round a number to a specific number of decimal places: - Look at the digit at the next decimal place (e.g., the fifth decimal place if rounding to four) - If that digit is 5 or more, round up the digit you are rounding to. - If it's less than 5, leave the targeted digit as it is. For instance, if the tangent of an angle calculates to a long decimal like 3.142857, and you need the answer to four decimal places, you focus on the first five digits: - Here, 3.142857 rounds to 3.1429, because the fifth digit (5) prompts a rounding up. This method of rounding ensures consistency and standardization in mathematical results and documents.