Problem 55 - Algebra and Trigonometry

Question

Find the area of a triangle with sides of length 10 and 22 and included angle \(10^{\circ} .\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 19.1 square units.

Step 1: Understanding the Task

We need to find the area of a triangle given two sides and the included angle. In this case, the sides are 10 and 22 units, and the included angle is \(10^{\circ}\).

Step 2: Identify the Formula

Use the formula for the area of a triangle involving two sides and the included angle: \( A = \frac{1}{2}ab \sin C \), where \( a \) and \( b \) are the sides and \( C \) is the included angle.

Step 3: Substitute the Values

Substitute the values into the formula. Here, \( a = 10 \), \( b = 22 \), and \( C = 10^{\circ} \). The formula becomes \( A = \frac{1}{2} \times 10 \times 22 \times \sin(10^{\circ}) \).

Step 4: Calculate the Sine of the Angle

Find \( \sin(10^{\circ}) \). Use a calculator to get \( \sin(10^{\circ}) \approx 0.1736 \).

Step 5: Compute the Area

Calculate the area using the previously substituted values: \[ A = \frac{1}{2} \times 10 \times 22 \times 0.1736 = 19.096.\] Therefore, the area of the triangle is approximately 19.1 square units.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. It is especially helpful in solving problems involving right-angled triangles, but its principles can also be applied to other types of triangles.
To understand trigonometry, visualizing triangles and the relationships between their components is key.

• Trigonometric ratios, such as sine, cosine, and tangent, are crucial tools. They relate angles to side lengths in triangles.
• The sine function, for example, relates an angle in a triangle to the ratio of the opposite side's length to the hypotenuse in a right triangle.
• These ratios can also be extended to non-right triangles through the Law of Sines and the Law of Cosines, allowing us to solve for unknown angles and sides.

Trigonometry has many real-world applications like engineering, physics, and even 3D modeling. Understanding these concepts helps in calculating the area in triangles when two sides and an included angle are known.

Triangle with Included Angle

A triangle with an included angle refers to a situation where two sides and the angle between them in a triangle are known.
This is quite different from a right triangle where one angle is specifically 90 degrees.

• An included angle offers a unique way of calculating the triangle’s area without needing to know all side lengths or the other angles.
• When given an included angle, it's possible to apply trigonometrical formulas like the one for the area: \( A = \frac{1}{2}ab \sin C \), where \(a\) and \(b\) are the known sides and \(C\) is the included angle.
• This method is especially useful for any scalene or obtuse triangles where not all angles are apparent.

Understanding how to use an included angle effectively can open up various methods for solving complex geometric problems and ensure accurate calculations.

Sine Function

The sine function is one of the fundamental concepts in trigonometry. It expresses the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
It's a ratio that reaches beyond right triangles and applies to any form of triangle using the Law of Sines.

• The function is written as \( \sin \theta \), where \( \theta \) is the angle.
• Computing the sine of an angle is crucial when calculating areas using angles, especially in non-right triangles where sine helps relate angle size to the spread between sides.
• An understanding of the sine values for various angles aids in solving many trigonometric problems without always resorting to a calculator.

In the original exercise, the sine of 10 degrees was used to find the area of a triangle with specific side measurements and an included angle, showing how powerful and versatile the sine function can be in trigonometric calculations.