Problem 20 - Trigonometry

Question

Find all solutions to each of the following triangles: \(B=34^{\circ}, b=4.2 \mathrm{~cm}, a=4.2 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The triangle has angles \(A=34^{\circ}\), \(B=34^{\circ}\), and \(C=112^{\circ}\); sides are \(a=b=4.2\) cm.

Step 1: Identify Triangle Type

We have a triangle with sides \(a = 4.2\) cm and \(b = 4.2\) cm, and angle \(B = 34^{\circ}\). Since \(a\) and \(b\) are equal, this is potentially an isosceles triangle.

Step 2: Use the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We have \( \frac{4.2}{\sin A} = \frac{4.2}{\sin 34^{\circ}} \). This simplifies to \( \sin A = \sin 34^{\circ} \), which means \( A = 34^{\circ} \).

Step 3: Check for Alternate Angle Possibility

The sine function has the property that \( \sin(\theta) = \sin(180^{\circ} - \theta) \). Therefore, another possibility for angle \(A\) could be \(A = 180^{\circ} - 34^{\circ} = 146^{\circ}\). However, if \(A = 146^{\circ}\), then the triangle angle sum would be more than \(180^{\circ}\), which is not possible. Hence, \(A = 34^{\circ}\) is the only possibility.

Step 4: Solve for Angle C

Use the triangle angle sum property \( A + B + C = 180^{\circ} \) to solve for \(C\). Substituting \(A = 34^{\circ}\) and \(B = 34^{\circ}\), we have \(34^{\circ} + 34^{\circ} + C = 180^{\circ}\). Hence, \(C = 180^{\circ} - 68^{\circ} = 112^{\circ}\).

Step 5: Summary of Triangle Solution

The solutions for the triangle are \(A = 34^{\circ}\), \(B = 34^{\circ}\), \(C = 112^{\circ}\), with equal sides \(a = b = 4.2\) cm.

Key Concepts

Isosceles Triangle

An isosceles triangle is a type of triangle that has at least two sides of equal length.
This characteristic makes isosceles triangles unique because they have two angles that are also equal. In our exercise, we have a triangle with sides \(a = 4.2\) cm and \(b = 4.2\) cm, making it an isosceles triangle.
Here is why isosceles triangles are important:

• Symmetry: An isosceles triangle is symmetrical along the axis that divides the triangle into two equal halves. This symmetry helps in various geometrical calculations and constructions.
• Angle Properties: If two sides are equal, the angles opposite those sides are equal. This simple rule makes calculating other parts of the triangle easier using laws and properties.
• Applications: Isosceles triangles are frequently used in architectural designs and other engineering fields for their stable nature.

If we identify a triangle as isosceles, it simplifies our calculations, as we only need to find the unknown properties of the triangle that are not symmetrical.

Law of Sines

The Law of Sines is a fundamental principle in trigonometry that is essential for solving triangles. It relates the sides of a triangle to its angles through the sine function.
The Law of Sines states:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] This law is particularly useful as it allows us to find unknown angles or sides of a triangle, especially when dealing with isosceles or scalene triangles where symmetry isn't enough.
In our exercise, since sides \(a\) and \(b\) are equal and \(B = 34^{\circ}\), we used this law to find \(A\) by:

• Setting up the equation: \(\frac{4.2}{\sin A} = \frac{4.2}{\sin 34^{\circ}}\) simplifies to \(\sin A = \sin 34^{\circ}\), giving us \(A = 34^{\circ}\).
• Understanding that \(\sin A = \sin B\) confirms the angles are equal, reinforcing that the triangle is isosceles.

The Law of Sines provides a straightforward method to solve triangles when direct measurement or simpler rules don't apply.

Angle Sum Property

The Angle Sum Property of a triangle is one of the basic, yet powerful, properties in geometry. It states that the sum of the interior angles in a triangle is always 180 degrees.
For any triangle with angles \(A\), \(B\), and \(C\), the rule can be written as:\[A + B + C = 180^{\circ}\]This property is invaluable for solving triangles because it provides a direct way to find any missing angle when the other two are known.
In our specific exercise, we used this property after determining that both \(A\) and \(B\) are \(34^{\circ}\) each. We then calculated \(C\) as follows:

• Started with the equation: \(34^{\circ} + 34^{\circ} + C = 180^{\circ}\).
• Solved for \(C\) by subtracting the sum of \(A\) and \(B\) from 180, resulting in \(C = 112^{\circ}\).

The Angle Sum Property provides a critical step in finding the remaining angle of a triangle when two angles are known, enabling us to solve for the complete triangle effortlessly.