Problem 1 - College Trigonometry

Question

Use the Law of Cosines to find the remaining side(s) and angle(s) if possible. $$ a=7, b=12, \gamma=59.3^{\circ} $$

Step-by-Step Solution

Answer

The side \(c\) is approximately 10.32 units long.

Step 1: Understanding the Problem

We are given two sides, \(a = 7\) and \(b = 12\), and the angle \(\gamma = 59.3^\circ\) across from the unknown side \(c\). We need to find the length of the side \(c\) using the Law of Cosines.

Step 2: Applying the Law of Cosines

The Law of Cosines states that \(c^2 = a^2 + b^2 - 2ab \cos(\gamma)\). Substitute the given values: \(c^2 = 7^2 + 12^2 - 2 \times 7 \times 12 \times \cos(59.3^\circ)\).

Step 3: Calculating \(c^2\)

Calculate the squares of \(a\) and \(b\): \(7^2 = 49\), \(12^2 = 144\). Thus, \(c^2 = 49 + 144 - 2 \times 7 \times 12 \times \cos(59.3^\circ)\).

Step 4: Finding \(\cos(59.3^\circ)\)

Use a calculator to find \(\cos(59.3^\circ) \approx 0.5150\).

Step 5: Substituting \(\cos(59.3^\circ)\) back

Replace \(\cos(59.3^\circ)\) with 0.5150 in the equation: \(c^2 = 49 + 144 - 2 \times 7 \times 12 \times 0.5150\).

Step 6: Simplifying the Equation

Calculate \(2 \times 7 \times 12 \times 0.5150 = 86.52\). So now \(c^2 = 193 - 86.52 = 106.48\).

Step 7: Calculating \(c\)

Find the square root of \(106.48\) to find \(c\). \(c \approx \sqrt{106.48} \approx 10.32\).

Step 8: Finding Remaining Angles Using Law of Sines (Optional)

If needed, use the Law of Sines to find another angle. For example, \(\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\). Use this to find angle \(\beta\) or \(\alpha\).

Key Concepts

Triangle Sides and Angles

Triangles have three sides and three angles, which can sometimes be found using various mathematical methods. In particular, knowing two sides and an included angle (the angle between these two sides) can help you find the unknown side or angles.

If we know the measures of two sides and the included angle, as in our exercise (where side lengths are 7 and 12, and the angle opposite the unknown side is 59.3 degrees), we can use the Law of Cosines. This helps us find the unknown third side and, subsequently, the remaining angles. This process involves using trigonometric laws, which relate the sides and angles of the triangle in specific ways.

The Law of Cosines formula is useful for finding the third side of a triangle, even in non-right triangles, and is a natural extension of the right-angle triangle's Pythagorean theorem.

Trigonometric Identities

Trigonometric identities play a crucial role in solving triangle problems, especially when non-right triangles are involved. The Law of Cosines, used in the given problem, is derived using trigonometric identities.

The formula is: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma) \] where \( c \) is the side opposite the angle \( \gamma \). This equation relates the sides of a triangle to the cosine of one of its angles, making it a powerful tool for solving triangles.

Cosine, a trigonometric function, is essential for these calculations. In our situation, \( \cos(59.3^\circ) \approx 0.5150 \), is a specific trigonometric value found using a calculator or trigonometric table.

This illustrates the importance of understanding trigonometric values and their roles in solving more complex mathematical problems involving triangle measurements.

Mathematical Problem Solving

Mathematical problem-solving involves applying formulas and logical reasoning to find answers. The solution to our triangle problem exemplifies this process in a structured approach.

Here's how you can tackle similar problems:

• Identify known and unknown variables (here, side \( c \) was unknown).
• Choose the appropriate formula, like the Law of Cosines, for the given data.
• Substitute the known values into the formula.
• Carry out the calculations to simplify the expressions and solve for the unknown.

Mathematics is about breaking problems into smaller steps. First, we calculated \( c^2 \), then took the square root to find \( c \). Each step must be clear and performed accurately to ensure correct results.

With practice, developing these problem-solving strategies becomes intuitive, allowing you to tackle similar mathematical challenges with confidence.