Problem 42 - Precalculus with Limits

Question

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(A\ =\ 5^{\circ}15'\), \(b\ =\ 4.5\), \(c\ =\ 22\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 0.8893 square units, after rounding to four decimal places.

Step 1: Convert the Angle

The angle is given in degrees and minutes. To make it suitable for calculation, it must be converted into decimal form. The conversion can be done using the fact that 1 degree = 60 minutes, hence 15 minutes = 15/60 = 0.25 degrees. Therefore, the angle \(A\) in decimal form is \(5.25^{\circ}\).

Step 2: Calculate Area using the Formula

Now, use the formula for the area of a triangle when two sides and the included angle are known, which is \(Area = 0.5 * b * c * \sin(A)\). Substitute the known values \(b = 4.5\), \(c = 22\), and \(A = 5.25^{\circ}\).

Step 3: Calculation

Remember to convert the angle A from degrees to radians before substituting into the sine function, as the sine function in most calculators operates in radians. The conversion can be done using the fact 1 degree = \(\pi/180\) radians. So, \(A = 5.25 * (\pi/180)\) radians. Solve the equation for the area after substitution: \(Area = 0.5 * 4.5 * 22 * \sin(5.25 * (\pi/180))\).

Key Concepts

Angle Conversion

Angle conversion is useful for understanding angles given in degrees and minutes. An angle might be presented as a combination like \(5^{\circ} 15'\). Here, \(5^{\circ}\) represents full degrees, while \(15'\) signifies minutes. Since 1 degree equals 60 minutes, this means:

• \(15' = 15/60\) degrees
• \(15' = 0.25\) degrees

This results in the numerical form \(5.25^{\circ}\) when combining \(5^{\circ}\) and \(0.25\) degrees, making further calculations manageable. Converting angles into a decimal form simplifies mathematical operations and ensures accuracy.

Trigonometry

Trigonometry helps solve problems involving triangles, especially when it comes to calculating areas and understanding angles between sides. By utilizing key trigonometric functions like sine, cosine, and tangent, we can explore the relationships between triangles' angles and their sides.

In our exercise's context, the sine function is particularly useful. We needed to determine the area of a triangle with two known sides and an included angle. The formula used in such a situation is:

• Area \(= \frac{1}{2} \cdot b \cdot c \cdot \sin(A)\)

This formula enables us to compute the triangle's area with ease once we have the appropriate angle and side lengths.

Sine Function

The sine function is pivotal for finding the relationship between a triangle's sides and angles. In right-angled triangles or problems involving non-right triangles, the sine function measures how much one side of a triangle is spread against the hypotenuse.

For any angle \(A\), the sine is defined as the ratio of the opposite side length to the hypotenuse in a right triangle. In the context of our problem, we are dealing with angle \(A = 5.25^{\circ}\). Thus, to use \(\sin(5.25^{\circ})\), the angle must first be converted from degrees to radians, as calculators often operate with the sine function in radian mode.

By finding the sine value in radians, we ensure precision in calculating the triangle's area.

Degrees to Radians Conversion

Degrees to radians conversion is vital in mathematics when working with trigonometric functions. While angles might initially be given in degrees, functions like sine, cosine, and tangent often require radian measures for computation. The conversion formula is:

• \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\)

In our exercise, the angle \(A = 5.25^{\circ}\) is converted into radians:

• \(5.25 \times \frac{\pi}{180} = A\) (in radians)

Using radians in trigonometric calculations aligns with the radian-based definition of trigonometric functions and is crucial for obtaining accurate results when using standard calculators or programming tools.