Problem 44 - Precalculus with Limits

Question

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(B\ =\ 84^{\circ}30'\), \(a\ =\ 16\), \(b\ =\ 20\)

Step-by-Step Solution

Answer

The area of the triangle is determined by substituting the given values into the formula and performing the calculation

Step 1: Convert angle from degree-minute format to decimal degrees

Given \(B = 84^{\circ}30'\). This is in degree-minute format. To convert it to decimal degree format, we divide the minutes by 60. Therefore, \( B = 84 + 30/60 = 84.5^{\circ} \)

Step 2: Substitute the values into the formula

Next, use \( a = 16 \) and \( b = 20 \) and the derived angle \( B = 84.5^{\circ} \) in the formula for the area of a triangle with sides and included angle: \[ Area = 0.5 * a * b * \sin(B) \]

Step 3: Compute for the Area

Substitute the values into the formula and calculate the result using a scientific calculator: \( Area = 0.5 * 16 * 20 * \sin(84.5) \). Round the answer to the nearest whole number if necessary

Key Concepts

Law of Sines

The Law of Sines is a fundamental concept in trigonometry, especially useful for solving problems involving triangles. This law offers a relationship between the lengths of the sides of a triangle and the sines of its angles. An easy way to state it is:

• \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]

where \( a, b, \) and \( c \) are the sides of the triangle, and \( A, B, \) and \( C \) are the opposite angles. In the context of triangle area calculation, instead of simply computing sides or angles, you can use this law to simplify finding unknown triangle parameters if you already know enough part of the triangle's sides and angles. This law helps create an easier path to understand relations in triangles, making it indispensable for any trigonometric problems that involve triangles with known sides and angles.

Trigonometry

Trigonometry is the branch of mathematics that studies relationships involving lengths and angles of triangles. It's foundational to understand numerous applications in math, science, and engineering. In our specific problem about finding the area of a triangle, trigonometry comes into play through the sine function. The sine of an angle in a right triangle is defined as the opposite side divided by the hypotenuse. However, in more general terms, it can be used in any type of triangle through functions like cosine and tangent.For our triangle:

• We employ the formula for the area using sine: \[ Area = 0.5 \times a \times b \times \sin(B) \]
• Here, \( a \) and \( b \) are sides, and \( B \) is the angle between them.

This use of trigonometry allows you to find the area even when the standard triangle formulas, like base times height, aren't applicable or straightforward to use.

Angle Conversion

Angle conversion is a crucial skill in trigonometry and geometry. Angles can be measured in degrees, minutes, seconds, or in decimal degrees. Understanding how to convert between these measurements helps in making computations smoother and more accurate.The angle in our problem is given in a degree-minute format as \( 84^{\circ}30'\). To convert it to decimal degrees:

• Divide the minutes by 60.
• Then, add this result to the degree value.
• So, \( 84^{\circ}30'\) becomes: \[ 84 + \frac{30}{60} = 84.5^{\circ} \]

Once you convert the angle into decimal degrees, you can directly use it for further calculations, such as using sine, cosine, or tangent functions in calculators. Proper angle conversion is especially helpful when working with non-standard angle measures, making it a necessary step in accurate trigonometric calculations.