Problem 35 - Algebra and Trigonometry

Question

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. $$ B=36^{\circ}, a=3 \text { yards, } c=6 \text { yards } $$

Step-by-Step Solution

Answer

The area of the triangle is approximately 5 square yards.

Step 1: Identify Given Measurements

Identify the given measurements from the exercise. Here angle B is \(36^{\circ}\), side a is 3 yards, and side c is 6 yards.

Step 2: Apply the Area Formula

Plug the given measurements into the formula for the area of the triangle. Therefore, you have A = \( \frac{1}{2} \cdot 3 \cdot 6 \cdot \sin 36^{\circ}\)

Step 3: Compute the Area

Perform the calculations. Start by calculating the sine of 36 degrees, then multiply the result by the side lengths and 0.5. Round your answer to the nearest whole number square unit if necessary.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics focused on the relationships between angles and sides in triangles. It introduces functions like sine, cosine, and tangent, which are ratios that uniquely correspond to the angles of a right triangle. These functions are vital in various fields like physics, engineering, and even in everyday calculations such as determining heights, distances, or geographical positioning.

Understanding trigonometry requires familiarity with right-angled triangles, as these are the basis of defining trigonometric functions. For any angle in a right triangle, the functions relate the angle to ratios of two sides of the triangle. This enables us to solve triangles when we know certain measurements — typically a combination of angles and side lengths. When it comes to calculating the area of a triangle, trigonometry offers formulas that use these functions, particularly when traditional methods like the 'base times height divided by two' are not applicable.

Sine Function

The sine function is one of the primary trigonometric functions, often abbreviated as 'sin.' In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the triangle's hypotenuse, the longest side. The definition extends to any angle, even beyond 90 degrees, using the unit circle approach for broader applications.

When calculating the area of a triangle where we don't have the height, but we do have two sides and the included angle (the angle between the two sides), the sine function becomes a hero. The formula for such a scenario is given by \( A = \frac{1}{2}ab\sin(C) \) where \( A \) is the area, \( a \) and \( b \) are the sides, and \( C \) is the included angle. This equation demonstrates how we transform geometric problems into trigonometric calculations, showcasing the sine function's crucial role.

Geometry

Geometry is the branch of mathematics that deals with the shapes, sizes, and properties of space configurations. It covers the study of points, lines, planes, angles, surfaces, and solids. In school, students often start with basic shapes like triangles, rectangles, and circles and gradually progress to more complex figures and three-dimensional objects.

In the context of our triangle area problem, geometry provides various formulas to compute areas, each tailored to the given dimensions and properties of the shape. The traditional formula for the area of a triangle, \( A = \frac{1}{2} \times base \times height \) is straightforward when the base and height are known. However, geometry is adaptable and offers alternative methods like using trigonometry when these measurements are not directly available. This flexibility allows us to solve for unknowns and understand the spatial relationships in our world.