Problem 9 - Algebra 2 and Trigonometry

Question

In \(8-13,\) find the exact value of the third side of each triangle. In \(\triangle P Q R, p=6, q=\sqrt{2},\) and \(\mathrm{m} \angle R=\frac{\pi}{4}\)

Step-by-Step Solution

Answer

The third side of triangle \(\triangle PQR\) is \(\sqrt{26}\).

Step 1: Understand the Given Variables

In the triangle \(\triangle PQR\), we have the following information:- \(p = 6\), which is one side of the triangle.- \(q = \sqrt{2}\), which is another side of the triangle.- \(\angle R = \frac{\pi}{4}\), which is an angle between sides \(p\) and \(q\). We are tasked with finding the third side, which we will call \(r\).

Step 2: Recall the Law of Cosines

The Law of Cosines is useful for solving triangles when you know two sides and the included angle. The formula is:\[ c^2 = a^2 + b^2 - 2ab \cos C \]Here, \(r\) will be the third side we are trying to find, \(p\) and \(q\) are the two given sides, and \(\angle R\) is the included angle C.

Step 3: Apply the Law of Cosines

Substitute the known values into the Law of Cosines formula:\[ r^2 = p^2 + q^2 - 2pq \cos(\angle R) \]Plug in the known quantities:\[ r^2 = 6^2 + (\sqrt{2})^2 - 2 \times 6 \times \sqrt{2} \times \cos\left(\frac{\pi}{4}\right) \]\(\cos\left(\frac{\pi}{4}\right)\) is known to be \(\frac{\sqrt{2}}{2}\).

Step 4: Calculate \(r^2\)

Compute each term:- \(6^2 = 36\)- \((\sqrt{2})^2 = 2\)- \(2 \times 6 \times \sqrt{2} = 12\sqrt{2}\)- \(12\sqrt{2} \times \frac{\sqrt{2}}{2} = 12 \times 1 = 12\)Use these to find \(r^2\):\[ r^2 = 36 + 2 - 12 = 26 \]

Step 5: Solve for \(r\)

Since \(r^2 = 26\), we take the square root of both sides to find \(r\):\[ r = \sqrt{26} \]

Step 6: Conclude the Exact Value of the Third Side

The exact value of the third side \(r\) of triangle \(\triangle PQR\) is \(\sqrt{26}\).

Key Concepts

Understanding Trigonometry

Trigonometry might sound a bit tricky, but it's merely a branch of mathematics that deals with the relationships between the angles and sides of triangles. The most used triangles in trigonometry are right-angled triangles. However, trigonometry is also handy for other types of triangles, like the one in our example, where we don’t have a right angle. Instead, we use different tools like the Law of Cosines to determine unknowns in a triangle.

Key tools of trigonometry:

• Sine, Cosine, and Tangent: These help in relating the angles to the sides of a right triangle. Even in non-right triangles, concepts like cosine are pivotal.
• Law of Sines and Cosines: Useful formulas that help us tackle non-right triangles by relating their angles and sides.
• Angle Measurement: In trigonometry, angles are often measured in radians rather than degrees. For example, \(\frac{\pi}{4}\) is a common angle measurement used.

Trigonometry provides these essential tools that allow us to explore and solve various real-world problems involving triangles.

Triangle Side Calculation

When it comes to calculating the sides of a triangle, it’s essential to know which method suits your triangle. In our exercise, knowing two sides and the measure of the included angle allowed us to use the Law of Cosines.

The Law of Cosines is as follows:\[c^2 = a^2 + b^2 - 2ab \cos C\]This formula helps find the third side of a triangle when you know two sides and the included angle. Here’s what you need to do:

• Identify the sides and the angle between them.
• Substitute these values into the Law of Cosines.
• Do the math to solve for the missing side.

The formula might seem complex if you're new to it, but with practice, pinpointing which values to substitute and solving it becomes more straightforward. Remember that understanding each element, like how the cosine of an angle is used, is crucial for the calculation.

Exploring Angle Measurement

In mathematics, angles can be measured in degrees or radians, with radians being more prevalent in trigonometry. The angle measurement is crucial as it impacts calculations, especially trigonometric ones like our exercise.

Key points about angle measurement:

• Degrees vs Radians: 360 degrees make a full circle, while 2π radians make the same circle.
• Conversion: To convert degrees to radians, multiply by \(\frac{\pi}{180}\). For radians to degrees, multiply by \(\frac{180}{\pi}\).
• Using in Formulas: Ensure you're consistent with angle measurements when using them in formulas like the Law of Cosines. For example, an angle given as \(\frac{\pi}{4}\) translates differently than an angle measured in degrees.

Understanding radians is particularly beneficial when solving problems involving periodic functions and waves, where angles appear naturally in radians.