Problem 14 - Algebra and Trigonometry

Question

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$ B=5^{\circ}, C=125^{\circ}, b=200 $$

Step-by-Step Solution

Answer

The remaining sides are \( a = 2306.7 \) units and \( c = 4788.9 \) units, and the remaining angle is \( A = 50 ^{\circ} \).

Step 1: Compute the remaining angle

Sum of all angles within a triangle totals 180 degrees. Since values for angles B and C are given, compute the third angle A using the formula: \( A = 180 - B - C \). Which will give \( A = 180 - 5 - 125 = 50 \) degrees.

Step 2: Use the Law of Sines to calculate other sides

Use the Law of Sines, i.e.,(a/sin A = b/sin B = c/sin C) where a, b, c are the sides opposite to the angles A, B, and C respectively. Here, we have the value for angle A and side b. So, firstly calculate side a using the formula: \( a = b \cdot \frac{sin A}{sin B} \). This gives \( a = 200 \cdot \frac{sin 50}{sin 5} = 2306.6 \), which to the nearest tenth is 2306.7. Secondly, calculate side c using the formula: \( c = b \cdot \frac{sin C}{sin B} \), yields \( c = 200 \cdot \frac{sin 125}{sin 5} = 4788.9 \) to the nearest tenth.

Key Concepts

law of sines

In the world of trigonometry, the Law of Sines is a powerful tool for solving triangles. Especially when you know some angles and sides, it helps fill in what you don’t know. Consider any triangle. This law states that the ratio of the length of a side to the sine of its opposite angle is constant across all three sides. You write it as:

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

Where:

• \(a, b, c\) are the sides opposite to angles \(A, B, C\) respectively.

It's very handy. Let's say you know two angles and one side, like in our example where \(B=5^{\circ}, C=125^{\circ}, b=200\). You can solve for the other sides \(a\) and \(c\) without needing to know angle \(A\) directly. Using the law, you substitute:

• For \(a\): \(a = b \cdot \frac{\sin A}{\sin B} = 200 \cdot \frac{\sin 50}{\sin 5} \)
• For \(c\): \(c = b \cdot \frac{\sin C}{\sin B} = 200 \cdot \frac{\sin 125}{\sin 5} \)

This approach finds \(a = 2306.7\) and \(c = 4788.9\). The Law of Sines makes connecting angles to sides straightforward in non-right triangles.

angle sum property

Every student exploring triangles must understand the angle sum property. It's a fundamental principle in trigonometry and geometry that the sum of the internal angles of a triangle is always 180 degrees. This simple rule helps you find unknown angles when two angles are given.

• To find the missing angle, subtract the total of the known angles from 180 degrees.

Consider the problem we looked at. We knew angles \(B = 5^{\circ}\) and \(C = 125^{\circ}\). Plug these into the angle sum rule:

• \(A = 180 - B - C = 180 - 5 - 125 = 50^{\circ}\)

Now you know angle \(A\). This step is quick and avoids any confusion. Using this property along with the Law of Sines helps seamlessly continue solving the entire triangle. It builds the base for further multi-step solutions.

triangle solving

Solving a triangle involves finding all its unknown sides and angles. The combination of the Law of Sines and the angle sum property equips you to tackle this task efficiently.

• Start by using the angle sum property to find any missing angles.
• Then, utilize the Law of Sines to solve for unknown sides or angles related to the remaining sides.

For our specific problem, the process went like this:

• Firstly, compute the missing angle \(A\) via the angle sum rule, receiving \(A = 50^{\circ}\).
• Next, solve for sides \(a\) and \(c\) through the Law of Sines. Here, sides are found using the sine ratios for angles \(A, B, C\).

Breaking down problems into these simple steps makes triangle solving straightforward. Knowing these techniques is essential, especially when working with non-right triangles. You practice applying these principles repeatedly, solidifying your understanding and skill in triangle-related problems.