Problem 51 - Contemporary Precalculus

Question

Given triangle \(A B C,\) with \(B=60^{\circ}, a=7,\) and \(c=15,\) solve the triangle as follows. (a) Show that \(b=13 . \text { [Hint: Example } 1 \text { of Section } 8.3 .]\) (b) Use the Law of sines to find angle \(C\). (c) Use the fact that the sum of the angles is \(180^{\circ}\) to find angle \(A\)

Step-by-Step Solution

Answer

Question: In triangle ABC, side a = 7, side c = 15, and angle B = 60°. Calculate the length of side b, and angles A and C. Answer: In triangle ABC, side b = 13, angle A ≈ 35.74°, and angle C ≈ 84.26°.

Step 1: Find Side b Using the Law of Cosines

Using the provided values (\(a=7\), \(c=15\), and \(B=60^{\circ}\)), we can apply the Law of Cosines to find the length of side \(b\). The Law of Cosines states that: \(b^2=a^2+c^2-2ac\cos{B}\). Plug in the given values and compute: \[b^2=7^2+15^2-2(7)(15)\cos{60^{\circ}}\] \[b^2=49+225-210(0.5)\] \[b^2=274-105\] \[b^2=169\] Next, we take the square root of both sides to find the length of side \(b\). \[b=\sqrt{169}\] \[b=13\] So, the length of side \(b\) is 13.

Step 2: Use the Law of Sines to Find Angle C

Now that we have found the length of side \(b\), we can use the Law of Sines to find angle \(C\). The Law of Sines states that \(\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\). We can use the following ratio to find angle \(C\): \[\frac{\sin{C}}{15}=\frac{\sin{60^{\circ}}}{13}\] Now, isolate \(\sin{C}\): \[\sin{C}=\frac{15\sin{60^{\circ}}}{13}\] Next, use a calculator to find \(\sin{C}\): \[\sin{C}=\frac{15(0.866)}{13}\] \[\sin{C}\approx 0.9971\] Now, find the angle \(C\) by taking the inverse sine of \(\sin{C}\): \[C=\sin^{-1}(0.9971)\] \[C \approx 84.26^{\circ}\] So, the angle \(C\) is approximately \(84.26^{\circ}\).

Step 3: Find Angle A Using the Angle Sum Property

Finally, we can find angle \(A\) using the fact that the sum of the angles of a triangle is \(180^{\circ}\). We have already found angles \(B\) and \(C\): \[A= 180^{\circ}-B-C\] \[A= 180^{\circ}-60^{\circ}-84.26^{\circ}\] \[A \approx 35.74^{\circ}\] So, the angle \(A\) is approximately \(35.74^{\circ}\). To summarize, in triangle \(ABC\): - Side \(b=13\) - Angle \(C \approx 84.26^{\circ}\) - Angle \(A \approx 35.74^{\circ}\)

Key Concepts

Law of Sines

Understanding the Law of Sines is like having the master key for unlocking the mysteries of any non-right triangle. Essentially, it's a way to relate the lengths of the sides of a triangle to the sines of its angles.

Here's the formula that makes the Law of Sines so powerful: \[\begin{equation}\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\end{equation}\]By using this principle, you can determine missing angles or sides as long as you know at least one angle-side pair and an additional angle or side.

When delving into the step-by-step solution for finding angle C, we applied this very law. After calculating side b using the Law of Cosines, we had enough information to set up our proportion based on the Law of Sines: \[\begin{equation}\frac{\sin C}{c} = \frac{\sin 60^\circ}{b}\end{equation}\]Calculating this gives us the sine of angle C, and from there, using the arcsine (inverse sine) function on a calculator provides us with the measure of angle C itself. This process showcases the Law of Sines as a vital tool for 'Solving Triangles,' which is another cornerstone of trigonometry.

Solving Triangles

Solving triangles is like completing a puzzle where each piece must fit perfectly. The objective here is to find all the missing sides and angles of a triangle when given some initial information. For non-right triangles, the Law of Sines and the Law of Cosines are our go-to strategies.

In the given exercise, we faced such a challenge. We started with side lengths a and c and angle B. Our first step was to find the missing side b using the Law of Cosines.

Remembering the Angles

Once side b was determined, we pivoted to the Law of Sines to find the elusive angle C. And to find angle A, we didn't need any more laws. We simply remembered that the sum of angles in any triangle adds up to 180 degrees. By subtracting the sum of angles B and C from 180, we solved the final piece of our triangle puzzle. This process demonstrates the seamless fusion of trigonometric principles to solve for unknowns in a triangle.

When solving triangles, it is essential to be systematic: find as many sides and angles as possible using the given information and then apply laws and properties like the Law of Sines (for ratios of sides to sines of opposite angles) and the sum of angles property (which states that the angles of a triangle always add to 180 degrees) to find the remaining unknowns.

Trigonometric Ratios

Trigonometric ratios are the bread and butter of solving triangles in trigonometry. They relate the sides of a right triangle to its angles, and extend to non-right triangles through laws like the Law of Sines and Law of Cosines.

These ratios—namely sine, cosine, and tangent—are defined as follows:

• The sine (sin) of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
• The cosine (cos) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• The tangent (tan) of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

While these definitions apply directly to right triangles, we apply them to all triangles using the Law of Sines and the Law of Cosines. For instance, in the exercise solution for angle C, we used the sine ratio as part of the Law of Sines to establish a relationship between the known angle B and the unknown angle C. This shows just how intertwined the trigonometric ratios are with the fundamental laws of trigonometry in solving for unknowns in various types of triangles.