Problem 18 - Contemporary Precalculus

Question

Solve the triangle. The Law of Cosines may be needed. $$a=30, b=40, A=30^{\circ}$$

Step-by-Step Solution

Answer

Question: Using the given information, find the side lengths and angle measures of the triangle: side lengths a = 30 and b = 40, and angle A = 30°. Answer: The completed triangle has side lengths a = 30, b = 40, and c ≈ 16.48, and angle measures A = 30°, B ≈ 19.47°, and C ≈ 130.53°.

Step 1: Find the length of side c using the Law of Cosines

Using the Law of Cosines, we can find the length of side c by plugging in the values for sides a, b, and angle A: $$c^2 = a^2 + b^2 - 2ab \cos A$$ Substitute the given values: $$c^2 = 30^2 + 40^2 - 2(30)(40) \cos(30^{\circ})$$

Step 2: Calculate the length of side c

Calculate the length of side c: $$c^2 = 900 + 1600 - 2400 \cos(30^{\circ})$$ $$c^2 = 2500 - 2400 \left(\frac{\sqrt{3}}{2}\right)$$ $$c \approx 16.48$$

Step 3: Find angle B using the Law of Sines

Use the Law of Sines to find angle B: $$\frac{\sin B}{b} = \frac{\sin A}{a}$$ Substitute the values and solve for B: $$\frac{\sin B}{40} = \frac{\sin(30^{\circ})}{30}$$ $$\sin B = \frac{40}{30} \sin(30^{\circ})$$ $$\sin B = \frac{2}{3}\left(\frac{1}{2}\right)$$ $$B = \arcsin\left(\frac{1}{3}\right) \approx 19.47^{\circ}$$

Step 4: Find angle C

Since the sum of angles in a triangle is \(180^{\circ}\), we can find angle C by subtracting angles A and B from \(180^{\circ}\): $$C = 180^{\circ} - A - B$$ $$C = 180^{\circ} - 30^{\circ} - 19.47^{\circ}$$ $$C \approx 130.53^{\circ}$$

Step 5: Present the completed triangle

The triangle with sides a, b, and c, and angles A, B, and C is now solved: $$a = 30, b = 40, c \approx 16.48$$ $$A = 30^{\circ}, B \approx 19.47^{\circ}, C \approx 130.53^{\circ}$$

Key Concepts

Triangle Solving

Solving a triangle means finding all the missing pieces of a puzzle. In this case, our puzzle is the missing side and angles of a triangle given some initial values. The triangle we are solving here is specified by two side lengths and one angle. It's helpful to remember:

• A triangle has three sides and three angles.
• The sum of the interior angles in any triangle is always 180°.
• To solve a triangle, you need to find any missing sides or angles using the given information.

In our example exercise, we have two sides, labeled as \(a = 30\) and \(b = 40\), and one angle, \(A = 30^{\circ}\). We need to find the third side, \(c\), and the other two angles, \(B\) and \(C\). By using both the Law of Cosines and the Law of Sines, we tackle this puzzle step by step.

Law of Sines

The Law of Sines is a powerful tool when working with triangles. It's especially helpful in scenarios like solving triangles where some sides and angles are missing. This law relates the sides of a triangle to the sines of their opposite angles. The formula is:

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

The Law of Sines helps us find unknown angles or sides when we know:

• At least one side length and its opposite angle.
• Another angle or side length.

In the step-by-step solution of the original exercise, we used the Law of Sines to find the angle \(B\). Given \(b = 40\), and \(A = 30^{\circ}\), we can solve for \(B\) using:

• \(\frac{\sin B}{40} = \frac{\sin 30^{\circ}}{30}\)
• This lets us solve for \(B = \arcsin\left(\frac{1}{3}\right) \approx 19.47^{\circ}\)

Knowing \(B\), we move closer to solving the entire triangle by piecing together the angles and sides.

Trigonometry

Trigonometry is the branch of mathematics that studies triangles and the relationships between their sides and angles. It's essential in solving triangle problems like the one in the exercise. We use trigonometric functions, such as sine, cosine, and tangent, to relate the angles and sides of a triangle.One of the key applications of trigonometry in triangle solving is through laws like the Law of Cosines and Law of Sines. These help calculate unknown sides and angles:

• Law of Cosines: Useful for finding a side of the triangle when you have two sides and the included angle. The formula is \(c^2 = a^2 + b^2 - 2ab \cos C\), which is an extension of the Pythagorean theorem.
• Law of Sines: Helpful for finding unknown angles or another side using known side-angle pairs.

By understanding how these functions work, you develop a toolkit for engaging with a wide range of triangle problems.