Problem 11 - 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 \(\triangle A B C, a=15, b=18,\) and \(\mathrm{m} \angle C=60\)

Step-by-Step Solution

Answer

Side \( c \approx 17 \), angle \( A \approx 49^\circ \), angle \( B \approx 71^\circ \).

Step 1: Identify Given Information

We are given that in triangle \( \triangle ABC \), side \( a = 15 \), side \( b = 18 \), and angle \( C = 60^\circ \). We need to find the measures of the remaining sides \( c \) and angles \( A \) and \( B \).

Step 2: Use Law of Cosines to Find Side c

To find side \( c \), we will use the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab \cos C \]Substitute given values:\[ c^2 = 15^2 + 18^2 - 2 \cdot 15 \cdot 18 \cdot \cos 60^\circ \]\[ c^2 = 225 + 324 - 2 \cdot 15 \cdot 18 \cdot 0.5 \]\[ c^2 = 225 + 324 - 270 \]\[ c^2 = 279 \]\[ c = \sqrt{279} \approx 16.7 \]Round to the nearest integer: \( c \approx 17 \).

Step 3: Use Law of Sines to Find Angle A

Next, we use the Law of Sines to find angle \( A \): \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]Substitute known values into the equation:\[ \frac{15}{\sin A} = \frac{17}{\sin 60^\circ} \]\[ \frac{15}{\sin A} = \frac{17}{\sqrt{3}/2} \]\[ \sin A = \frac{15 \times \sqrt{3}/2}{17} \]\[ \sin A = \frac{15 \sqrt{3}}{34} \]\[ A = \sin^{-1}\left(\frac{15 \sqrt{3}}{34}\right) \approx 49^\circ \].

Step 4: Calculate Angle B

Since the sum of angles in a triangle is \(180^\circ\), we can find angle \(B\) using:\[ B = 180^\circ - C - A \]\[ B = 180^\circ - 60^\circ - 49^\circ \]\[ B = 71^\circ \]

Step 5: Final Step: Compile the Solution

We have found that side \( c = 17 \), angle \( A = 49^\circ \), and angle \( B = 71^\circ \).

Key Concepts

Law of Cosines

The Law of Cosines is a very handy tool when working with triangles where we know an angle and the two sides surrounding it, or when we know all three sides. To understand how it functions, we can think of it as an extension of the Pythagorean theorem to non-right triangles. If you can recall, the Pythagorean theorem works only for right triangles. However, with a slight modification, the Law of Cosines can handle any type of triangle. This is invaluable in cases where we cannot directly apply the Law of Sines.

• The basic formula is: \[ c^2 = a^2 + b^2 - 2ab \cos C \] where \( c \) is the side opposite the angle \( C \).
• In our specific example, we plug in the known values for \( a \), \( b \), and \( \cos C \), leading us to calculate \( c \).

The Law of Cosines helps us solve for an unknown side when we know two sides and the included angle (an angle between two sides), or to find an angle when all side lengths are known. In the example, it allowed us to find side \( c \) when we were given sides \( a \) and \( b \), and angle \( C \) at 60 degrees.

Law of Sines

After using the Law of Cosines to find the side opposite the given angle, our task shifts to finding the remaining angles. This is where the Law of Sines becomes incredibly useful. The Law of Sines relates the sides of a triangle to the sines of its angles, making it a great tool for solving triangles when we need to find unknown angles or sides.

The formula for the Law of Sines is: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]It allows us to find unknowns using known angles and sides, whether they are angles or opposite sides. This works well when we know either two angles and one side or two sides and a non-included angle.

• We used it to find angle \( A \) by rearranging the formula to solve for \( \sin A \), using already known values for \( a \) and \( c \), and the sine of the given \( C \).
• This method can then give us precise results because it directly relates angles and sides through their respective sines.

Ultimately, by utilizing the formula properly, we can effectively find the unknown angles or sides of any triangle when initial conditions are provided.

Triangle Properties

Understanding triangle properties is key to navigating problems involving triangles. The most fundamental rule to always remember is that the sum of the angles in any triangle is always 180 degrees. This knowledge is pivotal for our calculations.

• When two angles and one side are known, this rule allows us to easily find the third angle by subtracting the known angle measures from 180 degrees.

In the context of our exercise, once we had found angle \( A \), and were already given angle \( C \), finding angle \( B \) was straightforward.

This adherence to basic triangle properties ensures that no mistakes are made when calculating unknown angles.

Moreover, knowing these inherent properties allows us to check our work. If the angle sum condition isn't satisfied, it signals a calculation error somewhere in our workings.

Understanding these simple properties can service not only as a direct calculation tool but also as a verification check on our workings.