Problem 41 - Precalculus with Limits

Question

Finding the Area of a Triangle In Exercises \(39-46\) find the area of the triangle having the indicated $$A=150^{\circ}, \quad b=8, \quad c=10$$

Step-by-Step Solution

Answer

The area of the triangle is 20 square units.

Step 1 Understand and arrange the given information

Here, we are given:Angle \(A = 150^{\circ}\), Side \(b=8\), Side \(c=10\). We're supposed to find the area of the triangle with these measures using the formula \(A = 0.5 * b * c * sin(A)\).

Step 2 Convert degrees to radians

The sine function in most calculators operates in radians, not degrees. So, we need to convert angle A from degrees to radians. 1 degree = 0.0174533 radiansSo, \(A = 150^{\circ} * 0.0174533 = 2.61799\) radians.

Step 3 Apply the formula

Now, we can substitute these values into the formula and solve for the area.\(A = 0.5 * b * c * sin(A) = 0.5 * 8 * 10 * sin(2.61799)\)

Step 4 Calculate the area

Carrying out the multiplication gives the area as \(A = 20\) square units.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It is especially useful in calculating unknown dimensions or angles when some elements of a triangle are already known. In practical terms, trigonometry often involves three main functions: sine, cosine, and tangent. These functions relate the angles of a triangle to the ratios of its sides. This makes it possible to find unknown measures in geometric and real-world applications, from architecture to engineering.

• Sine (sin): This function gives the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
• Cosine (cos): This function gives the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (tan): This function provides the ratio of the opposite side to the adjacent side.

Understanding these ratios and how they relate is at the core of solving triangular problems using trigonometry. It's a gateway to understand more complex concepts and application in fields such as calculus, physics, and even space travel!

Sine Function

The sine function is crucial for calculating the area of a triangle when you know two sides and the angle between them. The formula you use is: \[ A = \frac{1}{2} \times b \times c \times \sin(A) \]
Here, \(b\) and \(c\) are the lengths of the two sides, and \(A\) is the angle between these sides. The sine function translates this angle to a ratio that helps calculate the area more efficiently.

• For an angle measured in degrees, you need to ensure that your calculator is set to the degree mode.
• For a better understanding, remember: in a unit circle, the sine of an angle is the y-coordinate of the point where the corresponding ray intersects the circle.

Relative angles are often involved in more advanced trigonometric problems, but a good grip on the sine function is crucial for mastering them.

Radians Conversion

Understanding radians is crucial when dealing with the sine function in trigonometry, especially since the function often expects inputs in radians.
Radians are another way to measure angles, where a full circle is \(2\pi\) radians, which is approximately 6.28318. Therefore, converting degrees to radians is essential when calculations are needed on devices set to radian mode.

• Conversion formula: To convert from degrees to radians, multiply the degree value by \(\frac{\pi}{180}\).
• Example: For \(150^{\circ}\), the conversion is \(150 \times \frac{\pi}{180} = 2.61799\) radians.

This conversion ensures accuracy in trigonometric calculations since most scientific calculators default to radian mode. It's a small step but makes a big difference in keeping calculations consistent and reliable.