Problem 11 - Trigonometry

Question

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=120 \text { inches, } b=66 \text { inches, and } C=60^{\circ} \text {, find } c \text {. } $$

Step-by-Step Solution

Answer

The length of side \(c\) is approximately 104.08 inches.

Step 1: Understand the Law of Cosines

In order to find side \(c\) in triangle \(ABC\) with sides \(a\), \(b\), and angle \(C\), we employ the Law of Cosines. This law helps us to find a side length given two sides and the angle between them: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where \(a = 120\), \(b = 66\), and \(C = 60^{\circ}\).

Step 2: Substitute Known Values

Substitute the given values into the Law of Cosines formula. \[ c^2 = 120^2 + 66^2 - 2 \times 120 \times 66 \times \cos(60^{\circ}) \] Recall that \( \cos(60^{\circ}) = \frac{1}{2} \).

Step 3: Calculate Individual Components

Calculate each component:- \(120^2 = 14400\)- \(66^2 = 4356\)- \(2 \times 120 \times 66 \times \frac{1}{2} = 7920\)

Step 4: Solve for \(c^2\)

Combine the calculated values into the equation to solve for \(c^2\):\[ c^2 = 14400 + 4356 - 7920 \]Simplify it to get:\[ c^2 = 10836 \]

Step 5: Solve for \(c\)

To find the length of \(c\), take the square root of \(c^2\):\[ c = \sqrt{10836} \]Calculate this to find:\[ c \approx 104.08 \]

Key Concepts

Triangle Solutions

When solving triangles, especially when one or more sides or angles are unknown, different methods can be employed. In the case of triangle problems, solutions often depend on the information given. For triangles like the one mentioned, where two sides and an included angle are known, the Law of Cosines becomes a pivotal tool. Common methods for triangle solutions include:

• Law of Cosines: Useful for finding a missing side or angle when you know two sides and the included angle.
• Law of Sines: Used when dealing with cases having two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
• Right Triangle Trigonometry: Involves trigonometric ratios like sine, cosine, and tangent, typically used for 90-degree triangles.

Understanding the given information is key. By applying these methods, you can solve any triangle configuration. The crucial point is selecting a strategy that matches the given and needed measurements.

Angle-Side Relationships

In any triangle, there’s a critical relationship between the angles and the sides. This relationship guides us to understand which side is potentially the largest or smallest. Generally:

• The longest side of the triangle is opposite the largest angle.
• The shortest side is opposite the smallest angle.
• If sides are equal in length, the angles opposite to them must be equal too.

This concept is used when evaluating given triangle measurements: determine which side or angle of a triangle is the largest or smallest based on what you know about the other parts of the triangle. For instance, when two sides are equal, as in an isosceles triangle, the angles opposite those sides are also equal. This principle complements the Law of Cosines or Law of Sines by providing context for what the calculated numbers represent.

Trigonometric Identities

Trigonometric identities are fundamental tools in solving triangles and understanding their properties. These identities relate angles and sides in different ways, enabling the evaluation of triangles even with limited information. The primary trigonometric functions include sine, cosine, and tangent:

• Sine (sin): Represents the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine (cos): Represents the ratio of the adjacent side to the hypotenuse.
• Tangent (tan): Represents the ratio of the opposite side to the adjacent side.

For non-right triangles, the Law of Cosines and Law of Sines derive their power from these identities. For example, knowing that \( \cos(60^{\circ}) = \frac{1}{2} \) simplifies the problem-solving process tremendously when applying the Law of Cosines. Such identities not only aid computations but also provide insight into angle-side relationships within triangles, as seen in our original exercise.