Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Law of cosines to solve the triangle. $$C=101^{\circ}, \quad a=\frac{3}{8}, \quad b=\frac{3}{4}$$

Step-by-Step Solution

Answer

The lengths of the sides of the triangle are approximately a=0.375, b=0.75, and c=0.8992. The measures of the angles are A≈39.93 degrees, B≈39.06 degrees, and C=101 degrees.

Step 1: Use Law of Cosines to find the third side

The Law of Cosines states \(c^2 = a^2 + b^2 - 2ab \cos C\). Given C=101 degrees, a=3/8, and b=3/4, this can be substituted into the equation to solve for c: \(c^2 = (3/8)^2 + (3/4)^2 - 2 * (3/8) * (3/4) * \cos (101)\).

Step 2: Simplify equation to find c

After calculating, we find c squared equals approximately 0.8086. Taking the square root of this number gives c = 0.8992.

Step 3: Find the remaining angles

Now we use the Law of Sines or the Law of Cosines to find the remaining angles. For instance, by rearranging the Law of Cosines, we can write: \(A = \arccos((b^2 + c^2 - a^2) / (2bc))\). Substituting the known values gives us approximately 39.93 degrees for angle A. Similarly, we find angle B by using the equation for angle A, but replacing a with b and b with a. After calculation, angle B is approximately 39.06 degrees.

Key Concepts

Solving Triangles

When we talk about solving a triangle, it means finding all the unknown sides and angles of that triangle. For triangles with angles and sides that do not fit the simple right angle criteria, such as scalene or obtuse triangles, the Law of Cosines is particularly useful. It is a way to link the side lengths with the angles, providing a solution even when the triangle isn't right-angled.

The process involves using the known values, which could be side lengths and an included angle, to find out the missing side or angles. In the exercise given, we start by finding the third side, using the given angles and sides. Once we know this, we can proceed to calculate the other angles.

Remember, when solving triangles:

• Identify known sides and angles first.
• Choose the right formula, based on which elements are missing.
• Substitute the known values into the formula and solve for the unknowns.
• Double-check calculations to ensure accuracy.

This approach assures a comprehensive solution for any triangle configuration.

Trigonometry

Trigonometry deals extensively with triangles, particularly the relationships between their angles and sides. It's not just about knowing how to compute angles and sides; understanding the principles behind these computations gives you a deeper insight into how shapes work in space.

The Law of Cosines is one of the fundamental principles of trigonometry. It is useful in finding a missing side of a triangle when two sides and the included angle are known, or finding angles when all sides are known. This law is a generalization of the Pythagorean theorem and works for all types of triangles.

Applications of trigonometry span various fields:

• In navigation and engineering to model waves and oscillations.
• In physics, especially for understanding angular motion.
• In architecture and computer graphics for design and simulation.

By mastering trigonometric laws like the Law of Cosines, you enhance your problem-solving toolkit and open up pathways to figure out real-world problems with precision.

Angles and Sides

Understanding the relationship between angles and sides is crucial in any triangle problem. In a triangle, each side length contributes to determining the size of its opposite angle, which is the essence behind the Law of Cosines.

In our given triangle problem, we know two sides and an angle, and we find the third side through the Law of Cosines. Once all sides are known, the remaining angles can be determined. In such cases, the other laws, like the Law of Sines or remaining form of the Law of Cosines, are applied.

To solidify understanding, remember:

• The angle opposite the longest side is always the largest angle.
• Equilateral triangles have equal sides and angles, while isosceles triangles have two equal sides and angles.
• The sum of angles in any triangle always equals 180°.

Understanding these relationships helps in predicting and calculating the components of a triangle, proving that sometimes the form is as important as the function when it comes to geometry.