Problem 25 - Algebra 2 and Trigonometry

Question

In \(3-38,\) find each function value to four decimal places. $$ \tan 172^{\circ} 18^{\prime} $$

Step-by-Step Solution

Answer

The value of \(\tan 172^{ ext{°}} 18^{ ext{'}}\) is approximately -0.1667.

Step 1: Convert Minutes to Decimal Degrees

First, convert the angle from degrees and minutes to decimal degrees. Since 1 minute is equal to 1/60 of a degree, you can calculate the conversion as follows:Given: \(172^{ ext{°}} 18^{ ext{'}} = 172 + \frac{18}{60}\)\(= 172 + 0.3 = 172.3^{ ext{°}}\)

Step 2: Calculate the Tangent

Use a calculator to find the tangent of the angle in decimal degrees. Ensure that your calculator is set to degree mode.Calculate: \(\tan 172.3^{ ext{°}}\)

Step 3: Round to Four Decimal Places

After calculating \(\tan 172.3^{ ext{°}}\), round the result to four decimal places. Suppose the calculator gives: \(\tan 172.3^{ ext{°}} = -0.1667\)

Key Concepts

Tangent

Understanding the tangent function is essential in trigonometry, as it is one of the three primary trigonometric functions alongside sine and cosine. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side.

• For example, in a right triangle, if the length of the side opposite to the angle is "O" and the length of the side adjacent to the angle is "A", the tangent of the angle would be \( \tan(\theta) = \frac{O}{A} \).
• When dealing with angles beyond a right triangle, the tangent function can still be applied using this ratio concept, but with circular functions.
• The tangent is periodic with a period of 180° or \(\pi\) radians, meaning that \(\tan(\theta + 180^{\circ}) = \tan(\theta)\).

It's important to note that tangent values can be positive or negative, depending on the angle's position in the four quadrants of the coordinate system. In the given exercise, to find \( \tan 172.3^{\circ} \), locate the angle in the second quadrant, where the tangent is negative, as indicated by the result \( -0.1667 \). This reflects that at the angle of 172.3°, the tangent value is slightly below the x-axis in a unit circle representation.

Decimal Degrees

Working with angles in decimal degrees simplifies many trigonometric calculations. Angles are often given in degrees and minutes:

• Degrees are denoted by the symbol "°" and minutes by "'", where 60 minutes equal 1 degree.
• To convert an angle from degrees and minutes to a single decimal degree format, divide the number of minutes by 60. So, an angle of 172° 18' becomes \( 172 + \frac{18}{60} \) or \( 172.3^{\circ} \).

Using decimal degrees makes it easier to input angles into a calculator when performing trigonometric functions like tangent, sine, or cosine. It also provides a more streamlined approach for further mathematical operations, particularly when precision is important, as in our case of finding function values to four decimal places.

Rounding Numbers

In mathematics, rounding helps simplify numbers while maintaining a degree of accuracy that is acceptable for the problem at hand. Here, rounding to four decimal places means adjusting the number to have four digits after the decimal point:

• If the fifth digit is 5 or higher, round up the fourth digit. If it is 4 or lower, leave the fourth digit as is.
• This rule ensures consistency across all calculations and helps maintain precision in final results.

For instance, if you calculate \( \tan 172.3^{\circ} \) and obtain a result like -0.166659, you would round to -0.1667. This kind of rounding becomes crucial in scientific and engineering calculations, allowing for a level of precision that is both practical and meaningful without being unnecessarily cumbersome.