Problem 20 - 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 \(\triangle D E F, d=36, e=72,\) and \(\mathrm{m} \angle D=30\)

Step-by-Step Solution

Answer

Remaining parts: \(\angle E = 90^\circ\), \(\angle F = 60^\circ\), \(f = 62\).

Step 1: Use the Law of Sines

The Law of Sines states that \( \frac{d}{\sin D} = \frac{e}{\sin E} = \frac{f}{\sin F} \). We will use this to find \( \angle E \). Given \( \angle D = 30^\circ \), \( d = 36 \), and \( e = 72 \), substitute into the Law of Sines: \[ \frac{36}{\sin 30^\circ} = \frac{72}{\sin E} \]. Since \( \sin 30^\circ = \frac{1}{2} \), \[ \frac{36}{0.5} = \frac{72}{\sin E} \].

Step 2: Solve for Sine of Angle E

From the equation \( \frac{36}{0.5} = \frac{72}{\sin E} \), calculate \[ 72 = 72 \cdot \sin E \]. This simplifies to \( \sin E = 1 \). Since \( \sin E = 1 \), it means \( \angle E = 90^\circ \).

Step 3: Determine Remaining Angle F

Since the sum of all angles in a triangle is \( 180^\circ \), calculate \( \angle F \): \[ \angle F = 180^\circ - \angle D - \angle E = 180^\circ - 30^\circ - 90^\circ = 60^\circ \]. Thus, \( \angle F = 60^\circ \).

Step 4: Use the Law of Sines Again

We now use the Law of Sines to find side \( f \). From the Law of Sines, we have:\[ \frac{36}{\sin 30^\circ} = \frac{f}{\sin 60^\circ} \]. Knowing \( \sin 30^\circ = 0.5 \) and \( \sin 60^\circ = \frac{\sqrt{3}}{2} \), substitue to get:\[ \frac{36}{0.5} = \frac{f}{\frac{\sqrt{3}}{2}} \].

Step 5: Solve for Side f

From \( \frac{36}{0.5} = \frac{f}{\frac{\sqrt{3}}{2}} \), simplify to find \( f \):Calculate \( 72 = \frac{f \cdot 2}{\sqrt{3}} \),\[ f = 72 \cdot \frac{\sqrt{3}}{2} \]. e \( f = 72 \cdot 0.866 = 62.4 \), so latter the nearest integer is \( 62 \).

Key Concepts

Solving Triangles

Solving triangles involves determining the unknown sides and angles when certain initial specifications, such as side lengths and angles, are provided. This process is particularly useful in various fields, including navigation, physics, and engineering. In the case of triangle DEF, with side lengths and an angle known, the Law of Sines becomes an invaluable tool for solving the triangle.

• We start with identifying known values for sides and angles.
• Using these, we can apply trigonometric identities like the Law of Sines to find unknowns.
• The process continues until all angles and side lengths are determined.

In triangle DEF, given side lengths and an angle, we first calculated the unknown angles using trigonometric functions. Then, we proceeded to solve for the remaining side by strategically incorporating known values and leveraging the relationships defined by the Law of Sines.

Angle Measures

Understanding angle measures is crucial in solving triangles. Angles, measured in degrees, are the essence of defining geometric shapes such as triangles. A triangle's interior angles always sum to 180 degrees. For triangle DEF, we were given one angle, and had to calculate the others.

• Begin with any known angle and side combinations.
• Utilize the known angle to employ trigonometric functions, like sine, for further calculations.
• Remember that once two angles are known, the third angle is simply the remainder needed to reach 180 degrees.

In step 2, we calculated angle E using the equation \( \sin E = 1 \), finding it to be 90 degrees. Then, using the sum of angles, we easily computed angle F to be 60 degrees.

Trigonometric Functions

Trigonometric functions are essential mathematical tools used to solve triangles. These functions relate the angles of a triangle to the lengths of its sides. In solving triangle DEF, trigonometric functions allowed us to efficiently find unknowns.

• The Law of Sines is a critical function in solving non-right triangles.
• This law relates the ratio of side lengths to the sines of their opposite angles.
• In our example, this mathematical relationship helped calculate unknown angle E and side f.

To find side f, the calculation utilized the given sine values for known angles. The function helped translate side relationships into comprehensible numerical values. Understanding and applying these functions are key to solving and understanding triangle properties in geometry.