Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ P Q=12, P R=15, \mathrm{m} \angle R=100 $$

Step-by-Step Solution

Answer

One triangle is possible with angles \(100^\circ, 35^\circ,\) and \(45^\circ.\)

Step 1: Analyze the Given Information

We are given two side lengths of a triangle, \(PQ = 12\), \(PR = 15\), and the measure of angle \(\angle R = 100^\circ\). Our task is to find the number of triangles possible with this information and then calculate the measures of all three angles if a triangle exists.

Step 2: Apply the Law of Cosines

We use the Law of Cosines to find the length of the third side \(QR\):\[ QR^2 = PQ^2 + PR^2 - 2 \cdot PQ \cdot PR \cdot \cos(\angle R) \]Substituting the given values:\[ QR^2 = 12^2 + 15^2 - 2 \cdot 12 \cdot 15 \cdot \cos(100^\circ) \]Calculate the right-hand side to solve for \(QR\).

Step 3: Solve for QR Length

Calculating the equation:\[ QR^2 = 144 + 225 - 2 \cdot 12 \cdot 15 \cdot (-0.1736) \]\[ = 369 + 62.496 = 431.496 \]\[ QR = \sqrt{431.496} \approx 20.77 \]Thus, the length of side \(QR\) is approximately 20.77 units.

Step 4: Calculate Other Angles Using Law of Sines

Next, use the Law of Sines to find the measures of angles \(\angle P\) and \(\angle Q\):\( \frac{\sin(\angle Q)}{12} = \frac{\sin(100^\circ)}{20.77} \)\( \sin(\angle Q) = \frac{12 \cdot \sin(100^\circ)}{20.77} \) Calculate \(\angle Q\).

Step 5: Solve for \(\angle Q\)

Calculate \(\sin(\angle Q)\):\( \sin(\angle Q) = \frac{12 \cdot 0.9848}{20.77} \approx 0.568 \)Using inverse sine, \(\angle Q = \sin^{-1}(0.568) \approx 35^\circ\).

Step 6: Find \(\angle P\)

The sum of angles in a triangle is \(180^\circ\). Use this to find \(\angle P\):\(\angle P = 180^\circ - 100^\circ - 35^\circ = 45^\circ\).

Key Concepts

Triangle Angle Calculation

To calculate the angles in a triangle, one fundamental rule is the sum of all interior angles equals 180 degrees. Knowing this rule is crucial when you are given some, but not all, the angles of a triangle, as you can often deduce missing angles easily. In our example, we know \( \angle R = 100^{\circ} \). Once we find one of the other angles, let's say \( \angle Q \), calculating the last angle, \( \angle P \), involves simply subtracting the known angles from 180 degrees. Another helpful method for finding unknown angles is the Law of Sines, which we used in our problem. This law says that the ratio of the length of a side to the sine of its opposite angle is constant across all three sides of the triangle:

• \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \)

With these rules, once you have enough known values, calculating the unknown angles becomes straightforward. Just substitute the known side lengths and angles into the formula and solve the equation.

Triangle Side Calculation

Calculating the sides of a triangle can often involve the Pythagorean Theorem for right triangles or, in our problem, the Law of Cosines, which works for any triangle. The Law of Cosines is especially handy when you have two sides and an included angle, or when you need to determine the length of a side but you don't have a right triangle.In our example, with \( PQ = 12 \), \( PR = 15 \), and \( \angle R = 100^{\circ} \), we employed the Law of Cosines:

• \( c^2 = a^2 + b^2 - 2ab\cos(C) \)

This allowed us to find the length of \( QR \). After plugging in the values and solving, we found that \( QR \approx 20.77 \). This calculation tells us how the sides of the triangle stretch relative to one another, which is essential in understanding the triangle's overall shape.Whenever calculating side lengths in triangles, remember the power of the Law of Cosines and how it generalizes the relationship between sides and angles beyond right triangles. It’s a versatile tool for tackling a wide variety of geometric problems.

Solving Triangles

Solving a triangle means finding all its side lengths and angle measures. To do this effectively, you'll often employ a mix of the Law of Sines and the Law of Cosines, each tuned to help with specific scenarios. For cases where you know two sides and an included angle (like in our example), the Law of Cosines is the ideal first step to find the unknown side. Once that's determined, the Law of Sines comes into play to help find the remaining angles. Here's a quick guideline to solve triangles:

• Identify what you know: sides, angles, or both.
• Choose the appropriate method: Law of Sines or Law of Cosines.
• Solve for the missing angles and sides.

In our scenario, after determining all sides and one angle, we used a combination of the angle sum property and inverse trigonometric functions to find the final angle. By systematically applying these laws and properties, you can solve virtually any triangle. With practice, you'll become adept at quickly determining which tool to use and how to apply it, making triangle solving an intuitive and almost automatic process.