Question

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=69 \mathrm{~cm}, \mathrm{BC}=52 \mathrm{~cm}, \mathrm{AC}=49 \mathrm{~cm}\)

Short answer

Question: Given triangle ABC with side lengths AB = 69 cm, BC = 52 cm, and AC = 49 cm, find the angles A, B, and C. Answer: Use the Law of Cosines to first find angle C, opposite the longest side AB. Next, use the Law of Sines to find angle A. Lastly, use the angle sum of a triangle to find angle B. Once you have found angles A, B, and C, the solved triangle ABC can be presented with the discovered angles and given side lengths.

Step-by-step solution

Identify the longest side and its opposite angle

In this problem, AB is the longest side, so we will first find the angle opposite this side, which is angle C.

Use the Law of Cosines to find angle C

Using the Law of Cosines, we have: \(c^2 = a^2 + b^2 - 2ab \cdot \cos{C}\) where a = BC = 52 cm, b = AC = 49 cm, and c = AB = 69 cm. Plugging these values into the equation to find \(\cos{C}\), we have: \((69)^2 = (52)^2 + (49)^2 - 2(52)(49)\cdot \cos{C}\) Solve for \(\cos{C}\), and then use the inverse cosine function to find the angle C.

Use the Law of Sines to find angle A

Now that we have angle C, we can use the Law of Sines to find angle A: \(\frac{\sin{A}}{a} = \frac{\sin{C}}{c}\) Plug in the values for a and c, as well as the value of \(\sin{C}\) that you found in step 2. Solve for \(\sin{A}\), and then use the inverse sine function to find angle A.

Find angle B using angle sum of a triangle

Since the angles in a triangle must sum up to 180 degrees, we have: \(A + B + C = 180^\circ\) Plug in the known values for angles A and C and solve for angle B.

Present the solved triangle ABC

Once you have found angles A, B, and C, you can present the solution as: \(\triangle ABC\) with \(\angle A\), \(\angle B\), and \(\angle C\), and given side lengths AB, BC, and AC.

Key concepts

Triangle Solving

Triangle solving involves finding unknown angles and lengths using given measurements and specific mathematical methods. In the context of solving triangles, like \(\triangle \mathrm{ABC}\) where side lengths are given, we primarily rely on the Law of Cosines and the Law of Sines.
These laws help in finding unknown angles and sides. Here's a step-by-step approach in solving these problems:

• First, identify which angles or sides of the triangle are unknown.
• Decide which mathematical law or theorem is applicable.
• Apply this formula, process the numbers, and solve for the unknowns.

It's important to approach these problems methodically to ensure a complete, correct solution.

Mathematics for Engineers

Mathematics for engineers often involves mastering specific problem-solving techniques, like the Law of Cosines, which are found frequently in engineering problems. Engineers use geometry to assess structures and dynamics within various systems.
For example, when calculating forces acting on a structure or understanding the forces in a truss, knowing how to solve triangles is essential. By utilizing triangles and their properties:

• Engineering designs become more precise.
• Structural integrity can be verified.
• Complex machinery and components can be analyzed effectively.

This makes the mathematical techniques for solving triangles invaluable to real-world engineering challenges.

Law of Sines

The Law of Sines is used when you have one angle and its opposite side known. It helps find unknown angles or sides in a triangle, particularly when it's not a right triangle. The formula is:\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \]This formula shows the relationship of the angles and their opposite sides.
When used effectively, the Law of Sines helps to solve any unknown component of a triangle when dealing with non-right triangles.

• The formula uses the sin function, which is critical in trigonometry.
• It emphasizes the proportional relationship in a triangle.

Be sure to double-check calculations when using this law, especially in combination with other techniques, to ensure accuracy.

Angle Finding in Triangles

Finding an angle in a triangle when all side lengths are known is usually done with the Law of Cosines. For instance, consider \(\angle C\) which is found using the formula:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos C \]This allows us to solve for \(\cos C\) and then use an inverse cosine function to find the angle itself (
This is crucial for accuracy in applications involving shapes or trigonometry. Other tips include:

• Use known angles to simplify finding unknown angles.
• Remember that the sum of angles in a triangle is always 180 degrees.

With these steps, knowing specific methods like this bolsters your toolkit for finding angles efficiently and accurately.