Problem 10 - Algebra 2 and Trigonometry

Question

In \(9-14,\) find the area of each triangle to the nearest tenth. In \(\triangle A B C, a=326, c=157, \mathrm{m} \angle B=72\)

Step-by-Step Solution

Answer

The area of the triangle is approximately 24250.8 square units.

Step 1: Understand the problem and apply the formula

To find the area of triangle \( \triangle ABC \) with known sides \( a \) and \( c \) and angle \( \angle B \), we use the formula: \[ \text{Area} = \frac{1}{2}ac \sin B \]. Let's apply the given values to find the area.

Step 2: Substitute known values into the formula

Use \( a = 326 \), \( c = 157 \), and \( \angle B = 72^\circ \). Substitute these into the formula: \[ \text{Area} = \frac{1}{2} \times 326 \times 157 \times \sin(72^\circ) \].

Step 3: Calculate \( \sin 72^\circ \)

Find \( \sin 72^\circ \) using a calculator: \( \sin 72^\circ \approx 0.9511 \).

Step 4: Calculate the area

Plug \( \sin 72^\circ \approx 0.9511 \) back into the expression: \[ \text{Area} = \frac{1}{2} \times 326 \times 157 \times 0.9511 \].

Step 5: Perform the multiplication

Calculate \( \frac{1}{2} \times 326 \times 157 \times 0.9511 = 24250.7899\).

Step 6: Round the result to the nearest tenth

Round 24250.7899 to the nearest tenth, which gives 24250.8. Therefore, the area of the triangle is \( 24250.8 \).

Key Concepts

triangle area formula

The triangle area formula is a handy tool in geometry, especially when you know two sides and the included angle. Instead of relying only on base-length and height like with right triangles, this formula perfectly serves when dealing with non-right triangles.

For these situations, the formula is expressed as: \[ \text{Area} = \frac{1}{2}ac \sin B \].Here, \( a \) and \( c \) are sides of the triangle, and \( \angle B \) is the angle between these sides. This formula is especially practical because it accommodates for non-perpendicular angles, making it versatile for any triangle configuration you might encounter.

Understanding this formula is crucial for solving complex geometry problems efficiently.

sine function

The sine function is a fundamental part of trigonometry, closely associated with right-angled triangles and circular motion. In trigonometry, it helps relate the angles of a triangle to the lengths of its sides.

• The sine of an angle in a right triangle, \( \theta \), is calculated via the ratio: \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \).
• For angles not forming right triangles, such as those in our exercise, calculators or tables provide their sine values directly.

In our context, the sine of angle \( B \), \( \sin 72^\circ \), is approximately 0.9511. This value plays a critical role in calculating areas using the formula we discussed. It's important to get acquainted with basic trigonometric functions, as they are cornerstones of geometry and many physics concepts.

angle measurement

Measuring angles correctly is vital to all geometry calculations. Angles are traditionally measured in degrees, where a full circle contains 360 degrees. In our problem, the angle of interest, \( \angle B \), is 72 degrees.

Proper angle measurement ensures accurate calculations, particularly when using trigonometric functions like sine. Calculators often require angle input in degrees or radians, so it's essential to know which unit to use. Usually, exercises like ours use degrees, but being comfortable converting between the two units is beneficial.

Knowing how to measure angles and use the degree system effectively simplifies conducting various geometry tasks.

geometry calculations

Geometry calculations provide a way to deal with shapes and their properties comprehensively. In this context, calculating the area of a triangle involves understanding the relations between its sides and angles.

• Understanding how the triangle area formula works enables you to calculate efficiently, especially using trigonometric functions like sine.
• Accuracy in calculations requires correct substitutions and arithmetic operations, as we performed when multiplying side lengths and the sine function result.
• Rounding final answers is a common practice in geometry to match required precision, such as rounding to the nearest tenth.

Developing skills in geometry calculations helps solve real-life problems involving space, design, and construction, where precise measurements are vital.