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Chapter 6: Problem 67

How is the sine function used to find the area of an oblique triangle?

Short Answer

Expert verified
The sine function is utilized in finding the area of an oblique triangle by applying the formula: \( Area = 0.5 * a * b * \sin(C) \). This formula is used specifically when only the lengths of the sides and one of the included angles are known. The formula works because the sine of an angle provides a ratio that represents the proportion of the side opposite the angle to the hypotenuse.

Step by step solution

01

Understanding the concept

Take into consideration that an oblique triangle can be any triangle that does not have a right angle (90 degrees). However, the formula to find the area of an oblique triangle using the sine function proves useful, particularly when only the lengths of the sides and one of the included angles are known.
02

Understanding the Formula

The formula for finding the area of an oblique triangle with the sine function is as follows: \( Area = 0.5 * a * b * \sin(C) \) where a and b are the lengths of two sides of the triangle and C is the angle included between them.
03

Applying the Formula

Now, apply this formula to calculate the area of the triangle. Plug the lengths of the sides and the measure of the included angle into the formula, and then carry out the calculation.
04

Understanding the Logic of the Formula

The reason this formula works is because the sine of an angle in a triangle gives the ratio of the length of the side opposite the angle to the length of the hypotenuse. In a right triangle, the area is base * height / 2, but in an oblique triangle, the height becomes the other side times the sine of the included angle. The result is that the area of an oblique triangle can be directly calculated from its side lengths and the included angle between them.

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Key Concepts

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

Understanding the Sine Function
The sine function is a fundamental concept in trigonometry that helps us relate angles to side lengths in a triangle. Specifically, in a right triangle, the sine of an angle represents the ratio between the length of the side opposite to the angle and the hypotenuse.
  • For a given angle \( \theta \), the sine function is defined as: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
  • This relationship allows us to calculate unknown side lengths when one side and an angle are known.
In oblique triangles, we do not have a right angle. However, the sine function remains a crucial tool, particularly when we know two sides and the angle between them. Using the sine function, we can work out various properties of the triangle, including its area, which we'll explore further using the triangle area formula.
Exploring the Triangle Area Formula
To find the area of an oblique triangle, we use a special formula that involves the sine function. This is particularly useful when you know two sides of the triangle and the angle between them.The formula is given by:\[ \text{Area} = 0.5 \times a \times b \times \sin(C) \]where:
  • \(a\) and \(b\) are the lengths of two sides.
  • \(C\) is the angle between those two sides.
This formula is effective because it effectively adapts the concept of height in a right triangle to our oblique triangle. In a right triangle, the area is simply \( 0.5 \times \text{base} \times \text{height} \). However, in an oblique triangle:
  • The other side multiplied by the sine of the included angle gives a length analogous to the height in a right triangle.
By incorporating the sine function, this formula extends the basic principle of calculating area to triangles without right angles.
Delving into Trigonometry
Trigonometry is the branch of mathematics that investigates the relationships between the angles and sides of triangles. It is central to understanding many geometric and architectural calculations.
  • At its core, trigonometry helps us find distances and measures angles in various structures.
  • It comprises functions like sine, cosine, and tangent, each relating different sets of angles and sides in triangles.
On one hand, trigonometry can handle right-angled triangles precisely using basic formulas. On the other hand, with the help of identities and trigonometric functions like the sine function, it extends to oblique triangles, enabling precise calculations even without a 90-degree angle. This extension makes it invaluable for solving real-world problems involving indirect measurements and complex shapes.

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Most popular questions from this chapter

Convert each rectangular equation to a polar equation that expresses \(r\) in terms of \(\theta\) $$x^{2}+y^{2}=9$$

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=\sin \theta$$

Exercises \(112-114\) will help you prepare for the material covered in the next section. In each exercise, use a calculator to complete the table of coordinates. Where necessary, round to two decimal places. Then plot the resulting points, \((r, \theta),\) using a polar coordinate system. $$\begin{array}{c|c|c|c|c|c|c|c} \hline \boldsymbol{\theta} & \boldsymbol{0} & \frac{\pi}{6} & \frac{\pi}{3} & \frac{\pi}{2} & \frac{2 \pi}{3} & \frac{5 \pi}{6} & \pi \\ \hline r=1-\cos \theta & & & & & & \\ \hline \end{array}$$

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$\theta=\frac{\pi}{2}$$

The rectangular coordinates of a point are given. Find polar coordinates of each point. Express \(\theta\) in radians. $$(2,-2 \sqrt{3})$$
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