Problem 30 - Precalculus with Limits

Question

In Exercises 27-32, determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle. Then solve the triangle. \(a = 11\), \(b = 13\), \(c = 7\),

Step-by-Step Solution

Answer

The angles of the triangle are \(A\), \(B\), and \(C\), calculated using the Law of Cosines and the Law of Sines. The exact values will depend on the result of the calculations in each step.

Step 1: Use the Law of Cosines to find the first angle

Start by calculating the largest angle, which is opposite the longest side. The longest side here is b=13. So, to find the angle \(B\), use the equation: \[B = \arccos{\left(\frac{a^2 + c^2 - b^2}{2ac}\right)}\]. Substituting the given values will give the value of \(B\).

Step 2: Use the Law of Sines to find the second angle

Now that we have one angle, we can use the Law of Sines to find another. Let's find angle \(A\) using the equation: \[A = \arcsin{\left(\frac{a \cdot \sin{B}}{b}\right)}\]. Note that we use the longest side and corresponding angle (in our case, side b and angle B) in the denominator for the Law of Sines.

Step 3: Use the fact that all angles in a triangle sum to 180° to find the third angle

Remember that the sum of all angles in a triangle is 180°. Use this fact to find the third angle: \[C = 180° - A - B\]

Key Concepts

Triangle Solving

Triangle solving involves finding all the unknown sides and angles of a triangle. It’s a classic trigonometry problem where the goal is to solve for these components using given information. In this exercise, we are given three sides of a triangle:

• \(a = 11\)
• \(b = 13\)
• \(c = 7\)

With this information, our task is to determine the angles using the most suitable trigonometric rules.
To effectively solve triangles, it's essential to identify whether the given triangle is a right triangle or not, as this will determine the method you use. Here, we will utilize the Law of Cosines and the Law of Sines, which are particularly useful for non-right triangles.

Angle Calculation

Calculating angles in a triangle when all three sides are known involves a systematic approach:

• First, use the Law of Cosines to find the largest angle, which is opposite the longest side (here, side \(b=13\)). The formula is:\[ B = \arccos{\left(\frac{a^2 + c^2 - b^2}{2ac}\right)} \]
• In this scenario, we enter the values for sides \(a\), \(b\), and \(c\) into the formula, allowing us to calculate angle \(B\). The value of angle \(B\) helps solve for the remaining angles.
• Next, the Law of Sines aids in determining another angle, either \(A\) or \(C\), knowing that \(\sin{A}/a = \sin{B}/b\). This law is especially helpful when one angle and opposite side are known.
• Finally, utilize the property that the sum of angles in a triangle always equals 180°. Once two angles are known, the third angle is simply the difference:\[ C = 180° - A - B \]

Each of these steps guides you toward uncovering the complete angle set for the triangle, making solving such problems methodical and logical.

Trigonometry

Trigonometry is a branch of mathematics focusing on the relationships between the angles and sides of triangles. It includes functions like sine, cosine, and tangent, which are the building blocks of solving triangles.

• **Law of Sines:** This is used when we know either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). It’s formulated as \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
• **Law of Cosines:** Utilized when we know two sides and the included angle (SAS) or three sides (SSS). Written as: \( c^2 = a^2 + b^2 - 2ab \cos{C} \).

For non-right triangles, these laws are pivotal in calculating unknown angles and sides, expanding trigonometry's utility beyond simple right-angle problems. Mastery of these concepts is crucial for solving a variety of trigonometric applications and real-world problems.