Problem 3 - Precalculus with Calculus Previews

Question

Use the Law of Cosines to solve the triangle. $$ a=8, b=10, c=7 $$

Step-by-Step Solution

Answer

Angles are approximately 52.93°, 82.65°, and 44.42°.

Step 1: Review the Law of Cosines Formula

The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), and angles \(A\), \(B\), and \(C\) opposite those sides respectively, the formula is \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\). We can rearrange this to find \(C\): \(\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\).

Step 2: Calculate Angle C

Using the sides given, \(a = 8\), \(b = 10\), and \(c = 7\), substitute these values into the rearranged formula:\[\cos(C) = \frac{8^2 + 10^2 - 7^2}{2 \cdot 8 \cdot 10} = \frac{64 + 100 - 49}{160} = \frac{115}{160}=0.71875\]Calculate the angle \(C\) by taking the arccosine:\[C = \arccos(0.71875) \approx 44.42^\circ\].

Step 3: Use the Law of Cosines to find Angle A

Now solve for angle \(A\) using the formula \(a^2 = b^2 + c^2 - 2bc \cdot \cos(A)\). Rearrange to find \(\cos(A)\):\[\cos(A) = \frac{10^2 + 7^2 - 8^2}{2 \cdot 10 \cdot 7} = \frac{100 + 49 - 64}{140} = \frac{85}{140} = 0.6071\]Find angle \(A\) by calculating:\[A = \arccos(0.6071) \approx 52.93^\circ\].

Step 4: Find Angle B Using Angle Sum Property

With \(A\) and \(C\) found, use the fact that angles in a triangle sum to 180 degrees: \(B = 180^\circ - A - C\):\[B = 180^\circ - 52.93^\circ - 44.42^\circ = 82.65^\circ\].

Step 5: Summarize the Results

The angles of the triangle are approximately: \(A \approx 52.93^\circ\), \(B \approx 82.65^\circ\), and \(C \approx 44.42^\circ\).

Key Concepts

Understanding Angle Calculation

Calculating angles in a triangle is essential when you are navigating through geometry or trigonometry problems. Understanding how different parts of a triangle relate to each other can make this task much easier. One useful approach is using the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. This law is especially useful when you know the lengths of all three sides of a triangle, but need to find an angle.

• The formula for the Law of Cosines is: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \), where \( a \), \( b \), and \( c \) are the sides of the triangle, and \( C \) is the angle opposite the side \( c \).
• You can rearrange this formula to solve for \( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \).
• By taking the arccosine, you get the angle \( C \), allowing you to calculate its measure.

Knowing how to manipulate formulas and computational tools like a calculator can significantly simplify the angle calculation process.

Exploring Triangle Properties

Understanding triangle properties is crucial, as it forms the foundation of understanding the relationships within a triangle. Triangles possess unique properties that differentiate them from other geometric shapes.

• The sum of the internal angles in a triangle is always \( 180^\circ \).
• This means once you’ve calculated two angles, finding the third becomes a simple subtraction problem: \( B = 180^\circ - A - C \).
• Triangles with three different side lengths are known as scalene triangles, like the one in the exercise you are solving.

Understanding these properties helps you to identify which mathematical tool or formula to use in specific contexts, ensuring you get the correct results for each side or angle. This not only builds confidence but also improves problem-solving efficiency.

Dive into Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles. It’s a key tool not just in geometry, but also in fields like physics, engineering, and even art.

• Trigonometry uses specific functions namely sine, cosine, and tangent to relate the angles of a triangle to its side lengths.
• In the context of the Law of Cosines, the cosine function is utilized to find the measure of an angle when the side lengths are known.
• This approach is particularly useful in non-right-angled triangles, allowing for versatile problem-solving strategies.

Having a solid grasp of trigonometric principles provides a deeper understanding of how triangles work, enabling solutions to seemingly complex problems with elegance and precision. Emphasizing on trigonometric functions, their definitions, and how to manipulate them is central to mastering these topics.