Problem 35 - Precalculus

Question

Find the area of each triangle with measures given. $$a=4, c=7, \beta=27^{\circ}$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 6.356 square units.

Step 1: Understand the given information

We have a non-right triangle with sides and angle given as follows: \( a = 4 \), \( c = 7 \), and angle \( \beta = 27^{\circ} \). Our task is to find the area of this triangle using the provided information.

Step 2: Use the formula for the area of a triangle with two sides and the included angle

To find the area of the triangle, we can use the formula: \( \text{Area} = \frac{1}{2}ac\sin{\beta} \), where \( a \) and \( c \) are sides of the triangle, and \( \beta \) is the included angle.

Step 3: Substitute the given values

Substitute the values of \( a \), \( c \), and \( \beta \) into the formula: \[ \text{Area} = \frac{1}{2} \times 4 \times 7 \times \sin{27^{\circ}} \].

Step 4: Calculate the value of \( \sin{27^{\circ}} \)

Using a calculator, we find that \( \sin{27^{\circ}} \approx 0.454 \).

Step 5: Calculate the area

Now substitute this value back into the formula: \[ \text{Area} = \frac{1}{2} \times 4 \times 7 \times 0.454 \]. Calculate the product: \[ \text{Area} = 2 \times 7 \times 0.454 = 6.356 \].

Key Concepts

Non-right Triangle

A non-right triangle is a triangle that does not have a 90-degree angle. Unlike right triangles, which have one angle exactly equal to 90 degrees, non-right triangles have all angles less than 90 degrees or one angle greater than 90 degrees (known as an obtuse triangle).
This type of triangle can be categorized into acute and obtuse triangles based on the size of its angles:

• Acute Triangle: All angles are less than 90 degrees.
• Obtuse Triangle: One angle is greater than 90 degrees.

In our original problem, the triangle is classified as an acute triangle because the included angle \( \beta \) is 27°, much less than 90 degrees. Non-right triangles require different formulas for calculations as they do not work with the basic Pythagorean theorem used for right triangles.

Included Angle

The included angle is an important concept in triangle geometry. It refers to the angle formed between two given sides of a triangle. In triangle area calculations, particularly for non-right triangles, knowing the included angle can be very helpful.
For clarifying the practical application:

• An included angle is located between two known sides, such as \( a \) and \( c \) in our problem.
• This angle, denoted as \( \beta \), helps us use trigonometric functions to find relationships between side lengths and calculate the area.

In the given exercise, \( \beta = 27^{\circ} \) is the included angle between side \( a = 4 \) and side \( c = 7 \). Knowing this angle allows us to apply the specific trigonometric function needed to solve for the triangle's area.

Trigonometric Function

Trigonometric functions are essential tools in geometry, especially when working with triangles. In the context of calculating the area of a non-right triangle, we specifically rely on the sine (\( \sin \)) function.
The sine function helps us relate the size of the angle to the ratio of the side lengths opposite that angle:

• In our problem, \( \sin{\beta} \) where \( \beta = 27^{\circ} \), calculates how the height of the triangle (in relation to the base) contributes to the area.
• The formula used, \( \frac{1}{2}ac\sin{\beta} \), showcases how the sine of the included angle can directly affect the calculated area.

The sine value, approximately 0.454 for 27°, was accessed via a calculator and plays a crucial role in ensuring the accuracy of our area computation. This illustrates the power and utility of trigonometric functions in translating geometric complexities into measurable outcomes.