Problem 14 - 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 \(\triangle A B C, a=8, \mathrm{m} \angle B=35,\) and \(\mathrm{m} \angle C=55\)

Step-by-Step Solution

Answer

\( \text{m} \angle A = 90^\circ, b = 5, c = 7 \).

Step 1: Determine the unknown angle

To find the unknown angle in the triangle, we can use the fact that the sum of the angles in a triangle is always 180 degrees. We have angles B and C, so we can write the equation: \( \text{m} \angle A + \text{m} \angle B + \text{m} \angle C = 180^\circ \). Substitute the known angles: \( \text{m} \angle A + 35^\circ + 55^\circ = 180^\circ \). Simplify to find \( \text{m} \angle A \).

Step 2: Calculate Angle A

Solve the equation from Step 1: \( \text{m} \angle A = 180^\circ - 35^\circ - 55^\circ \). Thus, \( \text{m} \angle A = 90^\circ \). There is one unknown angle which is Angle A and it measures 90°.

Step 3: Use Law of Sines to find side b

Using the Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} \). We know \( a = 8, \angle A = 90^\circ, \angle B = 35^\circ \). Thus, \( \frac{8}{\sin 90^\circ} = \frac{b}{\sin 35^\circ} \). Solve for \( b \) to find its length.

Step 4: Calculate side b

Since \( \sin 90^\circ = 1 \), the equation becomes: \( 8 = \frac{b}{\sin 35^\circ} \). Solve for \( b \): \( b = 8 \cdot \sin 35^\circ \). Calculate using a calculator: \( b \approx 4.6 \). Round to nearest integer, making \( b = 5 \).

Step 5: Use Law of Sines to find side c

Now, use Law of Sines again for side c: \( \frac{a}{\sin A} = \frac{c}{\sin C} \). We know \( a = 8, \angle A = 90^\circ, \angle C = 55^\circ \). Replacing, \( \frac{8}{\sin 90^\circ} = \frac{c}{\sin 55^\circ} \). Solve for \( c \).

Step 6: Calculate side c

Because \( \sin 90^\circ = 1 \), simplify the equation: \( 8 = \frac{c}{\sin 55^\circ} \). Therefore, \( c = 8 \cdot \sin 55^\circ \). Compute using a calculator: \( c \approx 6.6 \). Round to nearest integer, making \( c = 7 \).

Step 7: Solution Found

The measures of the remaining parts of the triangle are: \( \text{m} \angle A = 90^\circ, b = 5, \text{and} \ c = 7 \).

Key Concepts

Law of Sines

The Law of Sines is a fundamental tool in triangle trigonometry that relates the sides of a triangle to the sines of its angles. This law is particularly useful in solving triangles when you know certain angles and one side.

In mathematical terms, the Law of Sines states that for any triangle \( \triangle ABC \):

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

Here, \(a\), \(b\), and \(c\) are the lengths of the sides opposite angles \(A\), \(B\), and \(C\) respectively. This means that the ratio of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles of the triangle.

Using the Law of Sines, if you are given two angles and one side, you can find the unknown sides of the triangle. For example, in our original exercise, knowing angles \(B\) and \(C\), as well as side \(a\) made it straightforward to determine sides \(b\) and \(c\). Calculators can help compute sines quickly, ensuring precise values for side calculations.

Angle Sum Property

The Angle Sum Property of a triangle is one of the simplest yet most important properties in trigonometry and geometry. It states that the sum of the interior angles of a triangle is always 180 degrees.

• \( \text{m} \angle A + \text{m} \angle B + \text{m} \angle C = 180^\circ \)

When solving a triangle, if you know two of its angles, finding the third angle becomes straightforward using this property.

In our exercise, since we knew \( \text{m} \angle B = 35^\circ \) and \( \text{m} \angle C = 55^\circ \), we simply subtracted their sum from 180 degrees to find \( \text{m} \angle A \). This resulted in \( \text{m} \angle A = 90^\circ \). Knowing this, along with at least one side, significantly simplifies the use of other mathematical tools to find missing sides or confirm right triangle properties.

Right Triangle

When it comes to triangles, the right triangle is one of the most fundamental shapes, with one angle always measuring precisely 90 degrees. This right angle creates unique properties that affect how we can calculate the triangle's sides and other angles.

• In a right triangle, the side opposite the right angle is known as the hypotenuse, which is always the longest side.
• Other sides are referred to as legs.

A right triangle often allows the use of specific trigonometric ratios, such as sine, cosine, and tangent, to determine unknown side lengths and angles.

In the original exercise, after calculating \( \text{m} \angle A \) as 90 degrees, it confirmed that the triangle was a right triangle. This information was crucial as it enabled direct application of the sin function and Law of Sines to solve for the remaining sides \(b\) and \(c\). Understanding whether a triangle is a right triangle is pivotal as it narrows down the methods to employ, making solutions more efficient.