Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. \(C=74.08^{\circ}, B=69.38^{\circ}, c=45.38\) meters

Step-by-Step Solution

Answer

Angle A is 36.54°, side a is approximately 28.1 meters, and side b is approximately 44.0 meters.

Step 1: Find Angle A

Since the sum of angles in any triangle is always 180 degrees, we can find angle A using the formula: \[ A = 180^{\circ} - (B + C) \]Substitute the known values:\[ A = 180^{\circ} - (69.38^{\circ} + 74.08^{\circ}) \]Calculate:\[ A = 180^{\circ} - 143.46^{\circ} \]\[ A = 36.54^{\circ} \]

Step 2: Use the Law of Sines to Find Side 'a'

The Law of Sines states:\[\frac{a}{\sin A} = \frac{c}{\sin C}\]Substitute the known values:\[\frac{a}{\sin 36.54^{\circ}} = \frac{45.38}{\sin 74.08^{\circ}}\]Calculate \(\sin 36.54^{\circ}\) and \(\sin 74.08^{\circ}\):\[\frac{a}{0.5965} = \frac{45.38}{0.9659} \]Solve for \(a\):\[ a = 0.5965 \times \frac{45.38}{0.9659} \]\[ a \approx 28.1 \text{ meters} \]

Step 3: Use the Law of Sines to Find Side 'b'

Using the Law of Sines again, we have:\[\frac{b}{\sin B} = \frac{c}{\sin C}\]Substitute the known values:\[\frac{b}{\sin 69.38^{\circ}} = \frac{45.38}{\sin 74.08^{\circ}}\]Calculate \(\sin 69.38^{\circ}\):\[\frac{b}{0.9365} = \frac{45.38}{0.9659} \]Solve for \(b\):\[ b = 0.9365 \times \frac{45.38}{0.9659} \]\[ b \approx 44.0 \text{ meters} \]

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry that helps solve triangles. It's especially useful for non-right triangles where at least one side and the corresponding angle are known. This law states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle. Mathematically, it is expressed as:

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

Here, \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle, while \(A\), \(B\), and \(C\) are the angles opposite those sides respectively.
To apply this law, you need either:

• Two angles and a side (AAS or ASA cases).
• Two sides and a non-included angle (SSA case).

For example, using this law, we can find unknown sides or angles, making it a versatile tool for triangle solving.

angle sum in a triangle

In any triangle, the sum of the interior angles is always 180 degrees. This is one of the fundamental properties that all triangles, regardless of their shapes and sizes, obey. Knowing this property allows us to find the third angle if two angles are already known. For instance, if you have a triangle where angles \(B\) and \(C\) are known, the angle \(A\) can be found using the expression:

• \(A = 180^{\circ} - (B + C)\)

In the given problem, angles \(B = 69.38^{\circ}\) and \(C = 74.08^{\circ}\) were provided. Thus, angle \(A\) was calculated as:

• \(A = 180^{\circ} - (69.38^{\circ} + 74.08^{\circ})\)
• \(A = 36.54^{\circ} \)

Knowing the sum of angles is essential for building the foundation in solving triangles accurately.

sine function

The sine function is a fundamental part of trigonometry and plays a vital role in the Law of Sines. It relates an angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. Sine values vary between -1 and 1 and are periodic, making it a cyclical function frequently used in wave-related phenomena.
For angle \(\theta\), sine is defined as:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

In the context of the problem, the sine values for angles \(36.54^{\circ}\), \(69.38^{\circ}\), and \(74.08^{\circ}\) were crucial. They facilitated solving for unknown side lengths \(a\) and \(b\) using these calculated sine values for the respective angles: \(\sin 36.54^{\circ} = 0.5965\), \(\sin 69.38^{\circ} = 0.9365\), and \(\sin 74.08^{\circ} = 0.9659\). Calculating accurate sine values is essential for using trigonometric identities and formulas effectively.

trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It includes various functions like sine, cosine, and tangent which relate angles to side ratios in right-angled triangles. Trigonometry is not just limited to geometry but extends to describe phenomena in physics, engineering, and even music.
Two widely used laws in trigonometry are the Law of Sines and the Law of Cosines, both of which are key in solving triangles. These laws allow the determination of unknown angles and sides in both right and non-right triangles from known values. In our example problem, the Law of Sines was used to find unknown side lengths following the angle sum rule. The calculations involved require a solid understanding of trigonometric concepts, reflecting why trigonometry is vital in scientific and practical applications.
Understanding trigonometry equips you with powerful tools for spatial reasoning and problem-solving in contexts beyond simple geometric figures.