Problem 55 - Precalculus with Limits

Question

Surveying A triangular parcel of ground has sides of lengths 725 feet, 650 feet, and 575 feet. Find the measure of the largest angle.

Step-by-Step Solution

Answer

The measure of the largest angle is approximately 101.37 degrees.

Step 1: Identify the largest side

The largest side of the triangle is 725 feet.

Step 2: Apply the law of cosines

The law of cosines says that for any triangle with sides of lengths a, b, and c, and an angle C opposite the side of length c, the following equation is true: \(c^2 = a^2 + b^2 - 2ab \cos(C)\). Here, c is the largest side, so \(725^2 = 650^2 + 575^2 - 2*650*575*\cos(C)\).

Step 3: Solve for the angle C

Isolate \(\cos(C)\) on one side of the equation by subtracting \(650^2\) and \(575^2\) from both sides, then dividing by \(-2*650*575\). Afterwards, use the inverse cosine function to find the measure of angle C: \(C = \cos^{-1}( \frac{ 650^2 + 575^2 - 725^2}{2*650*575} ) \). This gives the largest angle in radians, convert this to degrees by multiplying by \(\frac{180}{\pi}\).

Key Concepts

Triangle Properties

To understand the problem, let's look at some key properties of triangles. A triangle is a three-sided polygon, which means it consists of three edges and three vertices. The sum of the internal angles in any triangle is always 180 degrees, which is a fundamental property. This helps in calculating unknown angles when two are known.

• There are different types of triangles based on side lengths and angles: equilateral, isosceles, and scalene.
• An equilateral triangle has all sides equal and every angle measures 60 degrees.
• An isosceles triangle has two sides of equal length and two equal angles.
• A scalene triangle, like the one in our problem, has all sides of different lengths, leading to all angles being different.

In problems involving the Law of Cosines, we focus primarily on scalene triangles, and knowing the lengths of each side helps us determine the angles.

Angle Measurement

Measuring angles in a triangle is crucial to solving various geometry problems. In the problem at hand, we must find the largest angle using the Law of Cosines, which will give us the angle in radians. Often, angles in mathematics can be expressed in two units - degrees and radians.

• Degrees are more common in everyday use: a full circle is 360 degrees.
• Radians are used more in higher mathematics and physics: a full circle is \(2\pi\) radians.

To convert from radians to degrees, we use the conversion factor \(\frac{180}{\pi}\). By multiplying the radian measure of an angle by this factor, we can find the angle's measure in degrees, which can be easier to interpret in practical situations.

Inverse Trigonometric Functions

Inverse trigonometric functions are vital tools in solving for angles when given certain side measurements of a triangle. They "undo" what regular trigonometric functions do. In our context, especially the cosine function.

• The cosine function \(\cos\theta\) relates the angle of a triangle to the ratio of the length of the adjacent side to the hypotenuse.
• To find the angle itself, we use the inverse cosine function, denoted as \cos^{-1}\, which gives us the angle when the value of the cosine is known.

The Law of Cosines facilitated finding the cosine of the angle. Using the inverse cosine, we calculated the angle's measure, allowing us to complete the solution. By taking \(\cos^{-1}\) of the value obtained from the Law of Cosines, we arrive at the largest angle of the triangle. Thus, inverse trigonometric functions bridge the gap between side measurements and angle calculations.