Problem 11 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) approximate the remaining parts. $$\beta=71^{\circ} 51^{\prime}, \quad b=240.0$$

Step-by-Step Solution

Answer

\(\alpha = 18.15^{\circ}, \ a = 712.4, \ c = 751.2\)

Step 1: Convert the Angle

First, convert the given angle \(\beta\) from degrees and minutes into decimal format to make calculations easier. The angle \(\beta=71^{\circ} 51^{\prime}\) is converted by adding \(\frac{51}{60}\) degrees to the 71 degrees.\[\beta = 71 + \frac{51}{60} = 71.85^{\circ}\]

Step 2: Find Angle \(\alpha\)

Since triangle \(ABC\) is a right triangle with \(\gamma = 90^{\circ}\), the sum of \(\alpha\) and \(\beta\) is \(90^{\circ}\).\[\alpha = 90^{\circ} - \beta = 90^{\circ} - 71.85^{\circ} = 18.15^{\circ}\]

Step 3: Calculate Side \(c\)

Now, use the trigonometric function for cosine to calculate side \(c\) (adjacent to angle \(\beta\)). Cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse.\[\cos(\beta) = \frac{b}{c}\]Solve for \(c\):\[c = \frac{b}{\cos(\beta)} = \frac{240.0}{\cos(71.85^{\circ})} \approx 751.2\]

Step 4: Calculate Side \(a\)

Finally, use the trigonometric function for sine to calculate side \(a\) (opposite to angle \(\beta\)). Sine of an angle is the ratio of the opposite side to the hypotenuse.\[\sin(\beta) = \frac{a}{c}\]Solve for \(a\):\[a = c \times \sin(\beta) = 751.2 \times \sin(71.85^{\circ}) \approx 712.4\]

Key Concepts

Angle Conversion

To effectively work with angles in trigonometry, it's essential to convert them into a user-friendly format for calculations. Angles are often represented in degrees, minutes, and seconds. However, working in fractions of a degree can simplify calculations. In this exercise, the angle \(\beta\) is given as \(71^{\circ} 51^{\prime}\). To convert this into decimal degrees:

• Note that 1 degree equals 60 minutes. Therefore, 51 minutes is a fraction of a degree, calculated as \(\frac{51}{60}\).
• Add this fraction to the degrees: \[\beta = 71 + \frac{51}{60} = 71.85^{\circ}\]

This conversion is crucial as it transforms angles into a single value, making them suitable for mathematical calculations like those seen in trigonometry. After this conversion, finding other angles and working with trigonometric functions becomes straightforward.

Trigonometric Functions

Trigonometric functions are at the heart of right triangle trigonometry. In a right triangle, where one angle is \(90^{\circ}\), the other two angles still follow basic trigonometric relationships. Such functions help in calculating unknown sides of triangles when specific angles or sides are known. For triangle ABC:

• **Cosine** is utilized to calculate side \(c\). It is defined as the adjacent side over the hypotenuse. Here, \(\cos(\beta) = \frac{b}{c}\).
• Rearranging the formula gives \[c = \frac{b}{\cos(\beta)} = \frac{240.0}{\cos(71.85^{\circ})} \approx 751.2\]
• **Sine** is used to find side \(a\). Sine is defined as the opposite side over the hypotenuse, \(\sin(\beta) = \frac{a}{c}\).
• Solving for \(a\), \[a = c \times \sin(71.85^{\circ}) \approx 712.4\]

Understanding these functions is key as they provide a method for calculating unknown quantities in right triangles using known elements. Trigonometric function values can be easily found using calculators or trigonometric tables.

Triangle Side Calculation

Calculating the sides of a right triangle involves using known angles and sides along with trigonometric functions. In this context, the aim is to find sides \(a\) and \(c\) given side \(b\) and angle \(\beta\):

• First, after converting \(\beta\) to decimal, identify which trigonometric functions can apply. For sides of the triangle relative to angle \(\beta\), use cosine for the adjacent side and sine for the opposite side.
• For side \(c\): knowing side \(b\) as adjacent, use \[c = \frac{b}{\cos(71.85^{\circ})}\] to find \(c\), yielding approximately \(751.2\).
• For side \(a\): using the calculated \(c\), apply \[a = c \times \sin(71.85^{\circ})\] to find \(a\), approximately \(712.4\).

Through these calculations, it becomes evident how essential angles and trigonometric functions are in discovering unknown triangle sides, cementing their place as a cornerstone of geometry and practical mathematics.