Problem 17 - College Trigonometry

Question

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \beta=102^{\circ}, \gamma=35^{\circ}, b=16.75 $$

Step-by-Step Solution

Answer

The remaining angle \(\alpha\) is \(43^{\circ}\), side \(a\) is approximately \(11.89\), and side \(c\) is approximately \(10.12\).

Step 1: Determine the third angle

In any triangle, the sum of all angles is always \(180^{\circ}\). Therefore, we can use this property to find the third angle \(\alpha\). We know \(\beta = 102^{\circ}\) and \(\gamma = 35^{\circ}\). So, \(\alpha = 180^{\circ} - \beta - \gamma = 180^{\circ} - 102^{\circ} - 35^{\circ} = 43^{\circ}\).

Step 2: Use the Law of Sines to calculate side c

The Law of Sines states: \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \]. We know \(b = 16.75\), \(\beta = 102^{\circ}\), and \(\gamma = 35^{\circ}\). We can find side \(c\) as follows: \[ c = \frac{b \cdot \sin(\gamma)}{\sin(\beta)} = \frac{16.75 \cdot \sin(35^{\circ})}{\sin(102^{\circ})} \]. Calculating this gives \(c \approx 10.12\).

Step 3: Calculate side a using the Law of Sines

Using the Law of Sines again: \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \]. From Step 1, \(\alpha = 43^{\circ}\), so we find \(a\) as: \[ a = \frac{b \cdot \sin(\alpha)}{\sin(\beta)} = \frac{16.75 \cdot \sin(43^{\circ})}{\sin(102^{\circ})} \]. This calculation yields \(a \approx 11.89\).

Key Concepts

Law of Sines

The Law of Sines is a powerful tool in trigonometry for solving triangles. It provides a relationship between the angles and sides of a non-right triangle.

• This law states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of the triangle.
• The general formula is given by: \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\]This means a single piece of information can help solve others.

To use the Law of Sines, you need at least one side-angle opposite pair. With two angles, finding the third is simple, and you can calculate the unknown sides. In the solution above, the given side-angle pair \((\beta, b)\) allows us to find the other sides.

angle sum of triangles

One of the fundamental properties of triangles is that the sum of their internal angles always equals \(180^\circ\). This principle is crucial when solving triangles because it allows us to find a missing angle if two angles are known.

• For example, if you know two angles in a triangle, \(\beta\) and \(\gamma\) in our exercise, \(\beta = 102^\circ\) and \(\gamma = 35^\circ\), you can easily find \(\alpha\) by subtracting the sum of the known angles from \(180^\circ\).
• The equation used is: \[\alpha = 180^\circ - \beta - \gamma\]This simple property ensures that no matter how complex a triangle might look, calculating angles remains straightforward.

solving triangles

Solving a triangle involves finding unknown sides and angles with the known values. To do this, you can use a variety of trigonometric rules and theorems.

• Start by using the angle sum property to find any missing angles if only two are given, as knowing all three angles is vital.
• Next, apply the Law of Sines if you have an angle-side opposite pair to find unknown sides. In our example, after determining \(\alpha\), the missing side lengths \(a\) and \(c\) were calculated using the formula: \[\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \text{ and } \frac{c}{\sin(\gamma)} = \frac{b}{\sin(\beta)}\]

These strategies are vital whether dealing with an acute, obtuse, or right triangle. By piecing together the angles and sides methodically, any triangle can be solved simply and accurately.