Problem 28 - Technical Mathematics with Calculus

Question

The sides of a triangle are in the ratio \(2: 3: 4 .\) Find the cosine of the largest angle.

Step-by-Step Solution

Answer

\( \cos(C) = \frac{1}{4} \)

Step 1: Identify the Sides of the Triangle

Given the ratio of the sides of the triangle as 2:3:4, recognize that the longest side, which is opposite the largest angle, must be proportional to the 4 in the ratio. If we denote the sides as a, b, and c, with c being the longest, then a: b: c = 2:3:4.

Step 2: Apply the Law of Cosines

To find the cosine of the largest angle, which is opposite the longest side c, apply the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where C is the largest angle.

Step 3: Calculate the Cosine of the Largest Angle

Use the given ratios to express a and b in terms of c: a = (2/4)c and b = (3/4)c. Then substitute them into the Law of Cosines and solve for \cos(C): \( (4c)^2 = (2c)^2 + (3c)^2 - 2 \cdot (2c) \cdot (3c) \cos(C) \)

Step 4: Simplify and Solve for Cosine of C

After substituting the sides, the equation becomes \( 16c^2 = 4c^2 + 9c^2 - 12c^2 \cos(C) \), which simplifies to \( 16c^2 = 13c^2 - 12c^2 \cos(C) \). Solve for \cos(C) to get the cosine of the largest angle: \[-12c^2 \cos(C) = -3c^2 \] which gives us \( \cos(C) = \frac{-3c^2}{-12c^2} = \frac{1}{4}. \)

Key Concepts

Triangle Side Ratios

Understanding triangle side ratios is fundamental when dealing with problems in trigonometry, especially those that involve the Law of Cosines. The side ratios of a triangle can tell us much about its shape and the size of its angles. In a triangle with side lengths proportional to 2:3:4, we can infer that the longest side is opposite the largest angle.

In practice, when we're given that the sides of a triangle are in a specific ratio, we can assign variables to these sides while maintaining this ratio, say a,b, and c, where c is the longest side. If the sides' lengths are proportional to 2:3:4, it means any actual measurement of these sides will maintain this ratio, such as 2x, 3x, and 4x, where x is a common factor. This common factor is crucial in solving problems using the Law of Cosines, as it helps simplify calculations.

By understanding side ratios, and that they scale with a common factor, the complex relationship between the triangle's geometry and its angles becomes more apparent, enabling us to apply trigonometric laws more effectively.

Cosine of an Angle

The cosine of an angle in a triangle, specifically the angle opposite the longest side (the largest angle), is a key concept in trigonometry. Cosine is one of the primary trigonometric functions used to relate the angles of a triangle to the lengths of its sides.

In a right-angled triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. However, for non-right angled triangles, the cosine can be calculated for any angle using the Law of Cosines, which is fundamental in solving our example problem.

The Law of Cosines says that for any triangle with sides a, b, and c, and the angle C opposite side c, the relationship is given by: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] This formula is essential as it allows us to solve for the angle, given the sides' lengths, or conversely, to find a side's length given the angles and the other two sides. Understanding how to manipulate this equation is key to unlocking many trigonometry problems.

Trigonometric Identities

In order to solve a wide variety of trigonometry problems, knowledge of trigonometric identities is vital. These identities are equations that provide a relation between trigonometric functions and are true for all values of the involved variables. They can simplify complex expressions, convert one kind of function into another, and allow for the calculation of angles and lengths in geometric figures.

Some of the most important and basic identities are the Pythagorean identities, which relate the sine, cosine, and tangent of an angle. However, when it comes to the Law of Cosines, we depend on identities that link the sides of a triangle to the cosine of one of its angles.

The ability to recognize when and how to apply these identities often differentiates between a straightforward solution and a complex one. Mastering trigonometric identities is therefore an indispensable skill for students wishing to excel in trigonometry and practical applications such as physics, engineering, and surveying.