Problem 28 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \tan 266^{\circ} 27^{\prime} $$

Step-by-Step Solution

Answer

\(\tan 266^{\circ} 27' \approx 0.4856\).

Step 1: Convert Minutes to Degrees

The angle is given as \(266^{0}27'\). To deal with this, first convert the minutes to degrees. Since 1 degree = 60 minutes, 27 minutes is equivalent to \(\frac{27}{60}\) degrees. This gives \(266^{\circ} + \frac{27}{60} = 266.45^{\circ}\).

Step 2: Find the Tangent of the Angle

Next, calculate the tangent of the angle using a calculator set to degree mode. Enter \(266.45^{\circ}\) and press the tangent function (\(\tan\)). The result is approximately \(\tan(266.45^\circ) \approx 0.4856\).

Step 3: Round to Four Decimal Places

Ensure that the result \(0.4856\) is already rounded to four decimal places. Hence, there is no further rounding required.

Key Concepts

Angle Conversion

Understanding how to convert angles from one form to another is crucial in trigonometric calculations. Angles are often given in degrees, minutes, and seconds format. For example, an angle might be specified as \(266^{\circ}27'\). Here, 266 is the number of whole degrees, and 27 represents minutes. To convert this format into a decimal degree, which is more convenient for calculations, remember that 1 degree equals 60 minutes. This implies that any minute value must be divided by 60 to convert it to degrees. Therefore, to convert \(27'\) to degrees, compute \(\frac{27}{60} = 0.45\) degrees. By adding the converted value to the whole degree part, we arrive at \(266 + 0.45\), which equals \(266.45^{\circ}\). This simple conversion form allows you to proceed efficiently with calculations involving functions like sine, cosine, or tangent.

Trigonometric Functions

Trigonometric functions are a cornerstone of mathematics, particularly in fields like geometry and physics. They relate the angles and sides of triangles. The tangent function, denoted as \(\tan\), is one such function. It is defined in a right triangle as the ratio of the opposite side to the adjacent side but can also be applied to any angle reference when extended by the unit circle. To find the tangent of an angle, you typically use a calculator. Ensure that the calculator is set to degree mode when working with degrees. For instance, to find \(\tan(266.45^{\circ})\), enter the angle and then apply the tangent function. Using a calculator, we find \(\tan(266.45^{\circ}) \approx 0.4856\). The value arises from evaluating the ratio based on angle input in real numbers, allowing us to gain insights into the angle's trigonometric nature. Calculating the tangent is essential for solving various mathematical and real-world problems.

Rounding Decimals

Decimal rounding is a fundamental skill, especially when precision is required in mathematical calculations. Here, we're asked to round to four decimal places. This means keeping four numbers after the decimal point.Consider a calculated result like \(0.4856123\). To round to four decimal places, observe the fifth number past the decimal point, which in this example is a 2. When the fifth number is less than 5, the fourth number stays the same. Thus, \(0.4856123\) rounds to \(0.4856\). If the fifth number were 5 or greater, you would add 1 to the fourth decimal place. This level of rounding ensures that calculations remain accurate enough for most scientific and engineering applications while simplifying the number for practical use.