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Chapter 11: Problem 64

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the area of the triangle: \(a=14, b=11, C=30^{\circ}\)

Short Answer

Expert verified
The area of the triangle is 38.5 square units.

Step by step solution

01

Identify Given Values

Note the given values of the triangle: Side a = 14, Side b = 11, and Angle C = 30 degrees.
02

Use the Formula for Area

The area of a triangle with two sides and the included angle can be found using the formula: \[ \text{Area} = \frac{1}{2}ab\text{sin}C \]
03

Substitute the Values

Substitute the given values into the formula: \[ \text{Area} = \frac{1}{2} \times 14 \times 11 \times \text{sin}(30^\text{掳}) \]
04

Calculate the Sin Value

Know that \( \text{sin}(30^\text{掳}) = \frac{1}{2} \)
05

Simplify the Expression

Plug in the sin value and compute: \[ \text{Area} = \frac{1}{2} \times 14 \times 11 \times \frac{1}{2} = \frac{1}{2} \times 14 \times 11 \times 0.5 \]
06

Final Calculation

Perform the multiplication to find the area: \[ \text{Area} = \frac{1}{2} \times 77 = 38.5 \]

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Key Concepts

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

trigonometry
Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is essential in various fields such as engineering, physics, and architecture.

Some of the key functions in trigonometry include sine (sin), cosine (cos), and tangent (tan). These functions help in solving problems related to the dimensions and angles of triangles.

In this specific problem, we're using sine to find the area of a triangle. Understanding trigonometry is crucial as it gives us the tools to work with triangles in different scenarios.
triangle area formula
The triangle area formula involves using two sides of a triangle and the included angle between them. This formula is particularly useful when dealing with non-right-angled triangles.

The formula is as follows: \[ \text{Area} = \frac{1}{2}ab \text{sin}C \]

Here,
  • a and b are the lengths of two sides of the triangle
  • C is the included angle between those two sides
  • sin(C) represents the sine of angle C


By applying this formula, you can find the area of many types of triangles efficiently. It's a powerful method because it extends beyond the classic base-times-height approach.
sine function
The sine function is one of the primary trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse.

Mathematically, it's defined as: \[ \text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

For our problem, understanding that \[ \text{sin}(30^\circ) = \frac{1}{2} \] is crucial. The sine of 30 degrees simplifies many calculations and appears frequently in trigonometric problems.

Trigonometric values for common angles like 30掳, 45掳, and 60掳 are often memorized to speed up problem-solving. Mastery of these functions is fundamental for anyone studying mathematics or related disciplines.

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