Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-8,\) find the area of each \(\triangle A B C .\) $$ a=10, c=8, \sin B=\frac{3}{10} $$

Step-by-Step Solution

Answer

The area of the triangle is 12 square units.

Step 1: Identify Given Information

We are given a triangle \( \triangle ABC \) where \( a = 10 \), \( c = 8 \), and \( \sin B = \frac{3}{10} \). Our goal is to find the area of this triangle.

Step 2: Recall the Formula for Area of a Triangle with Sine

The formula for the area of a triangle using the sine of an angle is \( \text{Area} = \frac{1}{2}ab\sin C \). In our case, we know the sides \( a \), \( c \), and \( \sin B \). So, we can use \( \text{Area} = \frac{1}{2}ac\sin B \).

Step 3: Plug Values into the Formula

Substitute the known values into the formula: \( \text{Area} = \frac{1}{2} \times 10 \times 8 \times \frac{3}{10} \).

Step 4: Simplify the Expression

Calculate the expression: \[ \text{Area} = \frac{1}{2} \times 10 \times 8 \times \frac{3}{10} = 12 \].

Key Concepts

Understanding Trigonometry in Triangles

Trigonometry is the branch of mathematics concerned with the study of angles and the relationships between them. It's especially useful in dealing with triangles, where it helps you understand how to use angles and sides to find unknown properties. When working with triangles, trigonometric functions like sine, cosine, and tangent come into play.
The sine function, which is a key player in trigonometry, relates an angle to the opposite side and hypotenuse in a right triangle. In the context of non-right triangles, like triangle ABC, it helps us find the area by using the sine of one of its angles. This makes trigonometry a powerful tool in solving problems involving triangles, especially when certain sides and angles are known.

• Sine (\( \sin \theta \)) is crucial for finding area when angle and two sides are known.
• Learn the primary trigonometric functions: sine, cosine, and tangent.
• Apply trigonometry to solve real-world problems that involve triangles.

The Sine Rule Explained

The sine rule, also called the Law of Sines, is an equation that relates the sides of a triangle to the sines of its angles. This rule is very handy for figuring out unknown lengths or angles in a triangle when some elements are already known. The sine rule is given by:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]In the specific problem we tackled, the sine rule wasn't directly used to find lengths or angles, but its concept still underpins how we find the area using the sine of an angle.
When we use the area formula involving sine, we leverage the fact that any side of a triangle can be related to its opposite angle via the sine rule. This makes it possible to calculate the area even if the full picture of the triangle isn't known. Practically, this allows us to use the given \( \sin B \) in combination with sides \( a \) and \( c \) to determine the area effectively without needing a right-angle approach.

• The sine rule connects angles and sides in any triangle.
• Very useful in non-right triangles where traditional methods can't apply directly.
• Helps in finding unknown sides or angles using known ones and their sines.

Key Properties of Triangles

Triangles are one of the most fundamental shapes in geometry, with specific properties that apply no matter the type. Understanding these properties is key to solving problems associated with triangles, such as the one in our exercise.
Every triangle has three sides, angles, and vertices. The sum of its interior angles always equals 180 degrees, which is a constant property for all triangles. This property can be helpful in identifying unknown angles when one or two are known.
Additionally, the type of triangle is determined by its angles and sides. Common types include:

• Equilateral: All sides and angles are equal.
• Isosceles: Two sides and the angles opposite them are equal.
• Scalene: All sides and angles are different.

Understanding these properties allows you to apply correct formulas, such as the area calculation using sine. Knowing what type of triangle you are dealing with can also help in recognizing the best strategy for your calculations.