Problem 15 - Algebra 2 and Trigonometry

Question

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle D E F, d=72, e=48,\) and \(\mathrm{m} \angle F=110\)

Step-by-Step Solution

Answer

The measures are \( f = 104 \), \( E = 26^\circ \), and \( D = 44^\circ \).

Step 1: Identify Known Values

We have a triangle \( \triangle DEF \) with side lengths \( d = 72 \) and \( e = 48 \), and the measure of angle \( F = 110^\circ \). We need to find the remaining side \( f \) and the angles \( D \) and \( E \).

Step 2: Use the Law of Cosines

We can apply the Law of Cosines to find side \( f \). The formula is:\[f^2 = d^2 + e^2 - 2de \cos(F)\]Plug in the values:\[f^2 = 72^2 + 48^2 - 2(72)(48) \cos(110^\circ)\]Calculate \( \cos(110^\circ) \approx -0.3420 \) and continue:\[f^2 = 5184 + 2304 + 3301.77 \]\[f^2 = 10789.77\]\( f \approx \sqrt{10789.77} \approx 104 \).

Step 3: Use the Law of Sines to Find Angle E

Using the Law of Sines:\[ \frac{e}{\sin E} = \frac{f}{\sin F} \]\[ \sin E = \frac{e \sin F}{f} \]Substitute the known values:\[ \sin E = \frac{48 \sin 110^\circ}{104} \]Calculate \( \sin 110^\circ \approx 0.9397 \):\[ \sin E = \frac{48 \times 0.9397}{104} \approx 0.4334 \]Therefore, \( E \approx \arcsin(0.4334) \approx 26^\circ \).

Step 4: Find Angle D Using Angle Sum Property

The sum of angles in a triangle is \( 180^\circ \). Therefore, angle \( D \) can be found by:\[ D = 180^\circ - F - E \]\[ D = 180^\circ - 110^\circ - 26^\circ \]\( D \approx 44^\circ \).

Key Concepts

Law of Sines

The Law of Sines is a fundamental tool in trigonometry that helps solve triangles, especially useful in non-right triangles. It relates the sides of a triangle to the sines of its angles. The formula for the Law of Sines is given by:

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

Here, \(a\), \(b\), and \(c\) are the sides opposite angles \(A\), \(B\), and \(C\) respectively.
This law allows us to find unknown side lengths or angle measures when enough information is known. In the exercise with \(\triangle DEF\), after we figured out the side \(f\) using the Law of Cosines, we could apply the Law of Sines to find angle \(E\). This is done by setting up the equation:

• \( \frac{e}{\sin E} = \frac{f}{\sin F} \)

Substituting the known values into this equation allows us to solve for \(\sin E\) and find angle \(E\) using its arcsine.

triangle angle sum

The triangle angle sum property states that the sum of the internal angles in any triangle is always \(180^\circ\). This property holds true for all triangles, whether they are right, acute, or obtuse triangles.
In triangle problems, this rule allows you to find the third angle when two angles are known.
For example, if you know two angles in a triangle, subtracting their sum from \(180^\circ\) gives the measure of the third angle. In the example given, where \(\triangle DEF\) had angles \(F = 110^\circ\) and \(E = 26^\circ\), the third angle \(D\) was calculated using:

• \( D = 180^\circ - F - E \)

This provides a straightforward method to ensure all angles in a triangle are accounted for, an essential step in solving triangles.

calculate angle measure

Calculating unknown angle measures is a crucial part of solving triangles. To do this, we often use trigonometric laws and properties.
In the exercise example, the Law of Sines helps calculate angle \(E\) after the side \(f\) is known:

• The equation \( \sin E = \frac{48 \times 0.9397}{104} \approx 0.4334 \) was used.

This value is then used to find the angle:

• \( E \approx \arcsin(0.4334) \approx 26^\circ \)

To calculate angle \(D\), the triangle angle sum property is applied:

• \( D = 180^\circ - 110^\circ - 26^\circ \approx 44^\circ \)

These calculations allow us to find unknown angles efficiently and are straightforward with the right tools.

solve triangle sides

Determining the lengths of unknown triangle sides can be achieved using the Law of Cosines and the Law of Sines. These tools allow us to solve for sides effectively given sufficient initial information.
The Law of Cosines is particularly useful when you have two sides and the included angle. It can be used to find an unknown side. In the given exercise, it helped to find side \(f\) using:

• \( f^2 = d^2 + e^2 - 2de \cos(F) \)
• This becomes \( f^2 = 72^2 + 48^2 - 2 \times 72 \times 48 \times (-0.3420) \)
• Simplified to \( f^2 = 10789.77 \), leading to \( f \approx 104 \)

Once all side measures are calculated, the triangle is "solved," meaning all angles and sides are known, completing the puzzle of triangle properties.