Problem 16 - College Trigonometry

Question

Solve for the remaining side(s) and angle(s), if possible, using any appropriate technique. $$ \beta=7^{\circ}, \gamma=170^{\circ}, c=98.6 $$

Step-by-Step Solution

Step 1: Analyze the problem

Identify the type of problem and the appropriate mathematical technique to apply.

Step 2: Apply the technique and solve

Apply the identified mathematical method to obtain the solution.

Step 3: Verify the result

Check the answer by substitution or alternative methods to confirm correctness.

Key Concepts

Non-right Triangles

In trigonometry, triangles can vary greatly in their properties and classifications. A non-right triangle is one that does not have an angle of 90 degrees. This means that all the angles are either acute or obtuse. Non-right triangles are versatile, appearing often in practical applications and real-world problems. When working with non-right triangles, we employ laws such as the Law of Sines or the Law of Cosines, rather than relying solely on the simpler trigonometric ratios used for right triangles.

Non-right triangles are crucial in navigation, architecture, and physics, where right angles are not always present. For instance, the given triangle has angles \(\beta = 7^{\circ}\) and \(\gamma = 170^{\circ}\), indicating it is a very flat, obtuse triangle due to the presence of the large \(\gamma\) angle.

Triangle Angle Sum

One of the foundational principles in geometry is the triangle angle sum theorem. It states that the sum of all internal angles in a triangle is always 180 degrees. This theorem is instrumental when solving triangles because it allows us to find the unknown angle when two are given.

In the given problem, we are provided with two angles: \(\beta = 7^{\circ}\) and \(\gamma = 170^{\circ}\). To find the remaining angle, we simply subtract the sum of these angles from 180 degrees. Therefore, the third angle, \(\alpha\), can be calculated as follows:

\[ \alpha = 180^{\circ} - (\beta + \gamma) = 180^{\circ} - (7^{\circ} + 170^{\circ}) = 3^{\circ} \].

Knowing all angles in a triangle is essential before proceeding to solve for the sides.

Solving Triangles

Solving triangles involves finding unknown sides and angles, using known values either through trigonometric laws or basic arithmetic. Once the angle values (\(\beta = 7^{\circ}\), \(\gamma = 170^{\circ}\), \(\alpha = 3^{\circ}\)) are known in a non-right triangle, we can apply the Law of Sines:

\[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \]

This law relates the sides of a triangle to the sine of their respective opposite angles. In the context of the given exercise, the side \(c\) is known to be 98.6 units.

We can derive the other sides \(a\) and \(b\) by using the relationship:

\[ \frac{98.6}{\sin(170^{\circ})} = \frac{b}{\sin(7^{\circ})} \]

And similarly to find \(a\). This process highlights how linear relationships in triangles help simplify calculations and make solving feasible.

Angle-side Relationships

The relationship between angles and sides in any triangle is central to understanding trigonometry. In any triangle, the opposite side is directly related to the sine of the angle. Larger angles have larger opposite sides, an important feature when predicting relationships.

In the exercise, angle \(\gamma = 170^{\circ}\) is the largest angle, indicating that side \(c\) will be the longest. This is reflected in trigonometric identities and is evident in the application of the Law of Sines, which hinges on such relationships.

Knowing which side is largest can guide our expectations during solving, especially if you're working without exact numerical values and need to estimate proportionally. These relationships assist in sanity-checking your results, ensuring that computations align with geometric realities.