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Chapter 14: Problem 9

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle A B C, b=14.6, c=12.8, \mathrm{m} \angle A=56\)

Short Answer

Expert verified
The area of \(\triangle ABC\) is approximately 77.4 square units.

Step by step solution

01

Identify the Formula for Area of a Triangle

To find the area of a triangle when two sides and the included angle are given, use the formula: \ \[ \text{Area} = \frac{1}{2} \times b \times c \times \sin(A) \] where \( b \) and \( c \) are the lengths of the two sides, and \( A \) is the measure of the included angle.
02

Substitute Given Values into the Formula

We have \( b = 14.6 \), \( c = 12.8 \), and \( \angle A = 56^{\circ} \). Substitute these values into the area formula: \ \[ \text{Area} = \frac{1}{2} \times 14.6 \times 12.8 \times \sin(56^{\circ}) \]
03

Calculate the Sine of the Angle

Use a calculator to find \( \sin(56^{\circ}) \). The approximate value is: \ \[ \sin(56^{\circ}) \approx 0.8290 \]
04

Calculate the Area

Substitute \( \sin(56^{\circ}) \approx 0.8290 \) back into the area calculation: \ \[ \text{Area} = \frac{1}{2} \times 14.6 \times 12.8 \times 0.8290 \] \ First calculate \( 14.6 \times 12.8 = 186.88 \). \ Now, \( 186.88 \times 0.8290 = 154.86432 \). \ Finally, multiplying by \( \frac{1}{2} \), we get \( \frac{154.86432}{2} \approx 77.43216 \).
05

Round the Area to the Nearest Tenth

The area's calculated value was approximately 77.43216 square units. Rounding to the nearest tenth, the area of the triangle is \( 77.4 \) square units.

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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 part of trigonometry. It relates to angles in right-angled triangles. The sine of an angle is the ratio between the length of the side opposite the angle and the hypotenuse (the longest side of a right triangle). In mathematical terms, this is expressed as:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
However, the sine function's beauty lies in its application across all triangles when using the Law of Sines, or in our formula for finding the area when two sides and their included angle are known.
- **Unit Circle**: The sine function values can also be determined using the unit circle, where the circle's radius equals 1.- **Range**: The values of \(\sin(\theta)\) are between -1 and 1.- **Application**: It's used in various mathematical calculations beyond just triangles, including waves and oscillations in physics. This makes the sine function versatile and incredibly useful in geometry and beyond.
Exploring Triangle Properties
A triangle is a three-sided polygon, a fundamental shape in geometry, featuring some essential properties:
- **Sides**: A triangle has three sides, and the sum of any two sides is always greater than the third side. - **Angles**: The sum of the interior angles of a triangle is always 180 degrees. - **Area Calculation**: The area can be calculated in many ways, including using the base times height or trigonometric functions like the given exercise. - **Classification**: Triangles are classified by their sides (equilateral, isosceles, scalene) or their angles (acute, right, obtuse).
Understanding these properties helps in solving many geometric problems and assists in constructing real-world structures.
Basics of Trigonometry
Trigonometry is the branch of mathematics dealing with the study of triangles, specifically the relationships between their sides and the angles between these sides. It has several important concepts:
  • **Trigonometric Ratios**: Besides sine, there are other ratios like cosine and tangent that relate different sides of a right triangle to its angles.
  • **Connection to Geometry**: Trigonometry provides essential tools for solving geometric problems beyond simple triangles.
  • **Applications**: It's widely used in science, engineering, architecture, and navigation, due to its ability to relate angles and distances.
  • **Formulas**: Key formulas include the Pythagorean identity and the area of a triangle: \[ \text{Area} = \frac{1}{2} \times b \times c \times \sin(A) \]

These concepts are not just puzzle pieces in math problems but also cornerstones in fields that require precision and calculation.

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Most popular questions from this chapter

The length of one of the equal sides of an isosceles triangle measures 25.8 inches and each base angle measures 53 degrees. a. Find the measure of the base of the triangle to the nearest tenth. b. Find the perimeter of the triangle to the nearest inch.

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ c=18, b=10, \mathrm{m} \angle C=120 $$

Two points \(A\) and \(B\) are on the shoreline of Lake George. A surveyor is located at a third point \(C\) some distance from both points. The distance from \(A\) to \(C\) is 180.0 meters and the distance from \(B\) to \(C\) is 120.0 meters. The surveyor determines that the measure of \(\angle A C B\) is \(56.3^{\circ} .\) To the nearest tenth of a meter, what is the distance from \(A\) to \(B ?\)

When two forces act on an object, the resultant force is the single force that would have produced the same result. When the magnitudes of the two forces are represented by the lengths of two sides of a parallelogram, the resultant can be represented by the length of the diagonal of the parallelogram. If forces of 12 pounds and 18 pounds act at an angle of \(75^{\circ}\) , what is the magnitude of the resultant force to the nearest hundredth pound?

Aaron wants to draw \(\triangle A B C\) with \(A B=15\) inches, \(B C=8\) inches, and an area of 40 square inches. a. What must be the sine of \(\angle B ?\) b. Find, to the nearest tenth of a degree, the measure of \(\angle B .\) c. Is it possible for Aaron to draw two triangles that are not congruent to each other that satisfy the given conditions? Explain.
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