Problem 10 - Algebra 2 and Trigonometry

Question

In \(8-13,\) find the exact value of the third side of each triangle. In \(\triangle D E F, d=\sqrt{3}, e=5,\) and \(\mathrm{m} \angle F=\frac{\pi}{6}\)

Step-by-Step Solution

Answer

The exact value of the third side \(f\) is \(\sqrt{13}\).

Step 1: Identify the Formula Needed

To find the third side of a triangle when two sides and the included angle are known, we use the Law of Cosines. The Law of Cosines formula states: \[ f^2 = d^2 + e^2 - 2de \cos(F) \] where \(d\) and \(e\) are the known sides, and \(F\) is the included angle.

Step 2: Substitute Known Values

We know the values of \(d\), \(e\), and \(\angle F\): - \(d = \sqrt{3}\)- \(e = 5\)- \(\angle F = \frac{\pi}{6}\) Using these values, update the formula : \[ f^2 = (\sqrt{3})^2 + 5^2 - 2(\sqrt{3})(5)\cos\left(\frac{\pi}{6}\right) \]

Step 3: Calculate Each Term

First, calculate each individual term:- \((\sqrt{3})^2 = 3\)- \(5^2 = 25\)- Calculate \(\cos\left(\frac{\pi}{6}\right)\), which is \(\frac{\sqrt{3}}{2}\)- The product \(2(\sqrt{3})(5) = 10\sqrt{3}\)- Therefore, \(10\sqrt{3} \times \frac{\sqrt{3}}{2} = 15\)Now the equation looks like:\[ f^2 = 3 + 25 - 15 \]

Step 4: Simplify the Equation

By simplifying, we find:\[ f^2 = 3 + 25 - 15 = 13 \]

Step 5: Solve for \(f\)

To find \(f\), take the square root of both sides:\[ f = \sqrt{13} \]

Key Concepts

Triangle Sides Calculation

The calculation of the sides of a triangle, especially when two sides and an adjacent angle are given, is a classic problem in trigonometry. For triangles, where two sides and the included angle are known, we often use the Law of Cosines. This is an extension of the Pythagorean theorem. The formula allows us to find the length of the remaining side with preciseness.

Let's break down the formula step by step:

• The Law of Cosines states: \[ f^2 = d^2 + e^2 - 2de \cos(F) \]
• Here, \(d\) and \(e\) represent the lengths of the known sides of the triangle.
  
• \(\angle F\) is the included angle between these two sides.
  
• This formula helps calculate \(f\), the side opposite the angle \(\angle F\).
  

Taking our problem, where \(d = \sqrt{3}\), \(e = 5\), and \(F = \frac{\pi}{6}\), the formula becomes our guide to finding \(f\). All these elements blend into a straightforward calculation following a clear set of steps.

Included Angle

The included angle in a triangle is crucial when using the Law of Cosines. It is the angle formed directly between the two known sides. This angle is used to determine the magnitude of the missing side.

For our given triangle D E F, the included angle is \(\angle F\), which has a known measure:

• \(\angle F = \frac{\pi}{6}\) radians
  

When working with radians instead of degrees, remember that \(\pi\) radians equal 180 degrees. So, \(\frac{\pi}{6}\) is equivalent to 30 degrees. This conversion is critical for accurately applying trigonometry concepts, especially when using cosine functions, as these functions depend on the angle's measure.

Trigonometric Functions

Trigonometric functions are a core component of solving triangles, and the cosine function is particularly important in the Law of Cosines.

For our problem, the cosine of the included angle is needed:

• The cosine of \(\frac{\pi}{6}\) is \(\frac{\sqrt{3}}{2}\).
  

Understanding trigonometric values of common angles in radians helps streamline calculations in exercises like these. These trigonometric values are sometimes memorized due to their frequent use, and they play a vital role when substituting into formulas for problems involving angles measured in radians.

Exact Value

Finding the exact value of a side involves careful substitution and calculation in the Law of Cosines. For this particular triangle, we sought the length of side \(f\).

Our journey started by recognizing the need for a precise calculation without approximations. This is key to many math problems where exact values offer remarkable accuracy.

• First, we substituted our values into the equation, obtaining: \[ f^2 = 3 + 25 - 15 \]
• This simplifies to: \[ f^2 = 13 \]
• Taking the square root, we find: \[ f = \sqrt{13} \]
  This is the exact value needed, without rounding or decimal approximations.

Remember, achieving the exact value requires not only careful mathematical manipulation but also correct substitution of known values. This ensures precision and adds confidence in the result.