Problem 14 - Algebra and Trigonometry

Question

Solve triangle \(A B C\). \(b=60, \quad c=30, \quad \angle A=70^{\circ}\)

Step-by-Step Solution

Answer

By solving using the Law of Sines, angle B can be found, and subsequently angle C. Then side a is calculated using the same law.

Step 1: Apply the Law of Sines

To solve the triangle, we use the Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Plug in the known values: \( \frac{b}{\sin B} = \frac{60}{\sin B} \), \( \frac{c}{\sin C} = \frac{30}{\sin C} \), \( \frac{a}{\sin 70^{\circ}} \).

Step 2: Solve for Angle B

Calculate angle B using the Law of Sines: \( \sin B = \frac{b \cdot \sin A}{a} = \frac{60 \cdot \sin 70^{\circ}}{a} \). First, find angle C: \( \angle C = 180^{\circ} - 70^{\circ} - \angle B \). Use \( \angle A + \angle B + \angle C = 180^{\circ} \).

Step 3: Calculate Angle C

Once \( \angle B \) is determined, compute \( \angle C = 180^{\circ} - 70^{\circ} - \angle B \). Substitute values to find \( \angle C \) using triangle angle sum theorem.

Step 4: Determine Side a

Now use the relation \( \frac{a}{\sin 70^{\circ}} = \frac{60}{\sin B} \) to solve for \( a \). Rearrange to get \( a = 60 \cdot \frac{\sin 70^{\circ}}{\sin B} \).

Step 5: Verify and Conclude

Check all sides and angles using the Law of Sines and the fact that the sum of angles in the triangle is \(180^{\circ}\). Make sure there are no discrepancies and conclude the solution.

Key Concepts

Triangle Solving

Solving a triangle involves finding all the unknown sides and angles, given certain initial values. In the example of triangle \( ABC \), we start with two sides \( b = 60 \) and \( c = 30 \), and one angle \( \angle A = 70^{\circ} \). To solve this triangle, we need to determine:

• all the missing angles
• the unknown sides

Using the Law of Sines is essential here, as it establishes a relationship between the sides and angles of the triangle. This powerful trigonometric identity helps us find unknown values systematically. By carefully applying the steps, we ensure each component of the triangle is solved, providing a complete picture of its geometry.

Angle Calculation

Calculating angles in a triangle requires understanding the inherent property that the sum of angles in any triangle is always \(180^{\circ}\). This is fundamental when dealing with problems involving triangles. For triangle \( ABC \), we start with \( \angle A = 70^{\circ} \). To find \( \angle B \), we can use the Law of Sines:\[\sin B = \frac{b \cdot \sin A}{a}\]Once \( \angle B \) is determined, calculating the final angle \( \angle C \) is straightforward due to the angle sum property:\[\angle C = 180^{\circ} - \angle A - \angle B\] Finding these angles helps in fully solving the triangle, providing insight into its shape and orientation.

Trigonometric Identities

Trigonometric identities, including the Law of Sines, play a critical role in solving triangles. The Law of Sines states:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This equation creates a linkage between the sides and angles, allowing calculations to proceed smoothly. By substituting known values and systematically rearranging, you can solve for any unknown side or angle.Additionally, these identities also provide verification tools. After calculating all sides and angles, applying the identity checks consistency, ensuring that the solution respects both the trigonometric relationships and angle sum property. This confirmation step is crucial to validate findings and avoid errors.