/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 In Exercises 71-74, find the are... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 71

In Exercises 71-74, find the area of the triangle. $$ C=110^{\circ}, a=8, b=12 $$

Short Answer

Expert verified
The area of the triangle is approximately \(52.231\) square units.

Step by step solution

01

Understanding the Variables

In the problem, we are given that one angle \(C = 110^{\circ}\), and the sides \(a = 8\) and \(b = 12\). These will substitute into our formula to find the area of the triangle.
02

Substituting in the Formula

We now substitute the given values into the formula. We get \(A = \frac{1}{2} * 8 * 12 * \sin(110^{\circ})\). Now we have to calculate the value.
03

Calculating the Area

From our previous step, we can perform the calculation to get our final area. After calculating, we find that \(A = 52.231\) (rounded to three decimal places).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Area
The area of a triangle is a key concept in geometry, representing the space contained within the triangle's boundaries. Traditional methods for calculating this area rely on base and height measurements, as in the formula:
  • For triangles, the base and height can vary based on the orientation.
  • The simplest formula is: \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]
However, in scenarios where you don't have straightforward height, trigonometric formulas provide a valuable alternative. This comes particularly handy when dealing with non-right triangles or when given side-lengths and angles instead. By using trigonometric functions, we can find areas without needing to directly measure height, offering flexibility in diverse geometric problems.
Trigonometric Formula
Trigonometric formulas provide a universal way to solve problems involving triangles, especially when direct measurements like height aren't available. The most relevant formula to our triangle problem is the one for the area involving sine.
  • For any triangle with sides \(a\) and \(b\), and included angle \(C\), the area can be found with:\[A = \frac{1}{2} \times a \times b \times \sin(C)\]
This formula simplifies calculations significantly when you have two sides and the angle between them. Here's why it's so useful:
  • You don't need the height, just two sides and the sine of the included angle.
  • This makes it practical for non-right triangles that might appear in various geometrical or real-world scenarios.
The sine of the angle effectively scales the product of the two sides, adjusting for how the angle opens or closes the space within the triangle.
Sine Rule
The Sine Rule is a powerful tool in trigonometry, relating the sides of a triangle to the sines of its angles. While it's not directly used for the area calculation, it helps in understanding the interplay of angles and sides in a triangle. This rule states:
  • In any triangle, \[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]
Knowing the Sine Rule can help in determining missing angles or sides, which can then feed into area calculations.
  • It's especially beneficial when you have a mix of known and unknown quantities.
  • It underpins the understanding of non-right triangles and allows for complete triangle solving.
Combining the Sine Rule with the trigonometric formula for area gives a unified approach to tackling complex triangle problems. This interconnectedness is what makes trigonometry so powerful and widely applicable.

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