5.12: General Solutions of Triangles
While talking with your little sister one day, the conversation turns to shapes. Your sister is only in junior high school, so while she knows some things about right triangles, such as the Pythagorean Theorem, she doesn't know anything about other types of triangles. You show her an example of an oblique triangle by drawing this on a piece of paper:
Fascinated, she tells you that she knows how to calculate the area of a triangle using the familiar formula \begin{align*}\frac{1}{2}bh\end{align*} and the lengths of sides if the triangle is a right triangle, but that she can't use the formulas on the triangle you just drew.
"Do you know how to find the lengths of sides of the triangle and the area?" she asks.
Read on, and at the end of this Concept, you'll be able to answer your sister's question.
Watch This
James Sousa: The Law of Sines: The Basics
Guidance
In the previous sections we have discussed a number of methods for finding a missing side or angle in a triangle. Previously, we only knew how to do this in right triangles, but now we know how to find missing sides and angles in oblique triangles as well. By combining all of the methods we’ve learned up until this point, it is possible for us to find all missing sides and angles in any triangle we are given.
Below is a chart summarizing the triangle techniques that we have learned up to this point. This chart describes the type of triangle (either right or oblique), the given information, the appropriate technique to use, and what we can find using each technique.
Type of Triangle: | Given Information: | Technique: | What we can find: |
---|---|---|---|
Right | Two sides | Pythagorean Theorem | Third side |
Right | One angle and one side | Trigonometric ratios | Either of the other two sides |
Right | Two sides | Trigonometric ratios | Either of the other two angles |
Oblique | 2 angles and a non-included side (AAS) | Law of Sines | The other non-included side |
Oblique | 2 angles and the included side (ASA) | Law of Sines | Either of the non-included sides |
Oblique | 2 sides and the angle opposite one of those sides (SSA) – Ambiguous case | Law of Sines | The angle opposite the other side (can yield no, one, or two solutions) |
Oblique | 2 sides and the included angle (SAS) | Law of Cosines | The third side |
Oblique | 3 sides | Law of Cosines | Any of the three |
angles |
Example A
In \begin{align*}\triangle ABC, a = 12, b = 13, c = 8\end{align*}. Solve the triangle.
Solution: Since we are given all three sides in the triangle, we can use the Law of Cosines. Before we can solve the triangle, it is important to know what information we are missing. In this case, we do not know any of the angles, so we are solving for \begin{align*}\angle{A}, \angle{B}\end{align*}, and \begin{align*}\angle{C}\end{align*}. We will begin by finding \begin{align*}\angle{A}\end{align*}.
\begin{align*}12^2 & = 8^2 + 13^2 - 2(8)(13) \cos A \\ 144 & = 233 - 208 \cos A \\ - 89 & = - 208 \cos A \\ 0.4278846154 & = \cos A \\ 64.7 & \approx \angle{A}\end{align*}
Now, we will find \begin{align*}\angle{B}\end{align*} by using the Law of Cosines. Keep in mind that you can now also use the Law of Sines to find \begin{align*}\angle{B}\end{align*}. Use whatever method you feel more comfortable with.
\begin{align*}13^2 & = 8^2 + 12^2 - 2(8)(12) \cos B \\ 169 & = 208 - 192 \cos B \\ -39 & = -192 \cos B \\ 0.2031 & = \cos B \\ 78.3^\circ & \approx \angle{B}\end{align*}
We can now quickly find \begin{align*}\angle{C}\end{align*} by using the Triangle Sum Theorem, \begin{align*}180^\circ - 64.7^\circ - 78.3^\circ = 37^\circ\end{align*}
Example B
In triangle \begin{align*}DEF, d = 43, e = 37\end{align*}, and \begin{align*}\angle{F} = 124^\circ\end{align*}. Solve the triangle.
Solution: In this triangle, we have the SAS case because we know two sides and the included angle. This means that we can use the Law of Cosines to solve the triangle. In order to solve this triangle, we need to find side \begin{align*}f,\angle{D}\end{align*}, and \begin{align*}\angle{E}\end{align*}. First, we will need to find side \begin{align*}f\end{align*} using the Law of Cosines.
\begin{align*}f^2 & = 43^2 + 37^2 - 2(43)(37) \cos 124 \\ f^2 & = 4997.351819 \\ f& \approx 70.7\end{align*}
Now that we know \begin{align*}f\end{align*}, we know all three sides of the triangle. This means that we can use the Law of Cosines to find either \begin{align*}\angle D \end{align*} or \begin{align*}\angle E \end{align*}. We will find \begin{align*}\angle D \end{align*} first.
\begin{align*}43^2 & = 70.7^2 + 37^2 - 2(70.7)(37) \cos D \\ 1849 & = 6367.49 - 5231.8 \cos D \\ -4518.49 & = -5231.8 \cos D \\ 0.863658779 & = \cos D \\ 30.3^\circ & \approx \angle{D}\end{align*}
To find \begin{align*}\angle E\end{align*}, we need only to use the Triangle Sum Theorem, \begin{align*}\angle{E} = 180 - (124 + 30.3) = 25.7^\circ\end{align*}.
Example C
In triangle \begin{align*}ABC, A = 43^\circ, B = 82^\circ\end{align*}, and \begin{align*}c = 10.3\end{align*}. Solve the triangle.
Solution: This is an example of the ASA case, which means that we can use the Law of Sines to solve the triangle. In order to use the Law of Sines, we must first know \begin{align*}\angle C\end{align*}, which we can find using the Triangle Sum Theorem, \begin{align*}\angle{C} = 180^\circ - (43^\circ + 82^\circ) = 55^\circ\end{align*}.
Now that we know \begin{align*}\angle{C}\end{align*}, we can use the Law of Sines to find either side \begin{align*}a\end{align*} or side \begin{align*}b\end{align*}.
\begin{align*}\frac{\sin 55}{10.3} & = \frac{\sin 43}{a} && \frac{\sin 55}{10.3} = \frac{\sin 82}{b} \\ a & = \frac{10.3 \sin 43}{\sin 55} && \qquad \ b = \frac{10.3 \sin 82}{\sin 55} \\ a & = 8.6 && \qquad \ b = 12.5\end{align*}
Vocabulary
Law of Cosines: The law of cosines is an equation relating the length of one side of a triangle to the lengths of the other two sides and the sine of the angle included between the other two sides.
Law of Sines: The law of sines is an equation relating the sine of an interior angle of a triangle divided by the side opposite that angle to a different interior angle of the same triangle divided by the side opposite that second angle.
Guided Practice
1. Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of the triangle and label the given information. Also, state how many solutions (if any) the triangle would have. If a triangle has no solution or two solutions, explain why.
\begin{align*}A = 69^\circ, B = 12^\circ, a = 22.3\end{align*}, find \begin{align*}b\end{align*}
2. Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of the triangle and label the given information. Also, state how many solutions (if any) the triangle would have. If a triangle has no solution or two solutions, explain why.
\begin{align*}a = 1.4, b = 2.3, C = 58^\circ\end{align*}, find \begin{align*}c\end{align*}.
3. Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of the triangle and label the given information. Also, state how many solutions (if any) the triangle would have. If a triangle has no solution or two solutions, explain why.
\begin{align*}a = 3.3, b = 6.1, c = 4.8\end{align*}, find \begin{align*}A\end{align*}.
Solutions:
1. AAS, Law of Sines, one solution
2. SAS, Law of Cosines, one solution
3. SSS, Law of Cosines, one solution
Concept Problem Solution
Since you know that two of the angles are \begin{align*}23^\circ\end{align*} and \begin{align*}28^\circ\end{align*}, the third angle in the triangle must be \begin{align*}180^\circ - 23^\circ - 28^\circ = 129^\circ\end{align*}. Using these angles and the knowledge that one of the sides has a length of 4, you can solve for the lengths of the other two sides using the Law of Sines:
\begin{align*} \frac{\sin A}{a} = \frac{\sin B}{b}\\ \frac{\sin 23^\circ}{a} = \frac{\sin 129^\circ}{4}\\ a = \frac{4\sin 23^\circ}{\sin 129^\circ} = \frac{1.56}{.777}\\ a \approx 2\\ \end{align*}
And repeating the process for the third side:
\begin{align*} \frac{\sin A}{a} = \frac{\sin C}{c}\\ \frac{\sin 23^\circ}{2} = \frac{\sin 28^\circ}{c}\\ c = \frac{2\sin 28^\circ}{\sin 23^\circ} = \frac{.939}{.781}\\ c \approx 1.2\\ \end{align*}
Now you know all three angles and all three sides. You can use Heron's formula or the alternative formula for the area of a triangle to find the area:
\begin{align*} K = \frac{1}{2}bc\sin A\\ K = \frac{1}{2}(4)(1.2)\sin 23^\circ\\ K = \frac{1}{2}(4)(1.2)(.391)\\ K \approx .9384 \end{align*}
Practice
Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of the triangle and label the given information. Also, state how many solutions (if any) the triangle would have.
- \begin{align*}a=3, b=4, C=71^\circ\end{align*}, find \begin{align*}c\end{align*}.
- \begin{align*}a=8, b=7, c=9\end{align*}, find \begin{align*}A\end{align*}.
- \begin{align*}A=135^\circ, B=12^\circ, c=100\end{align*}, find \begin{align*}a\end{align*}.
- \begin{align*}a=12, b=10, A=80^\circ\end{align*}, find \begin{align*}c\end{align*}.
- \begin{align*}A=50^\circ, B=87^\circ, a=13\end{align*}, find \begin{align*}b\end{align*}.
- In \begin{align*}\triangle ABC\end{align*}, \begin{align*}a=15, b=19, c=20\end{align*}. Solve the triangle.
- In \begin{align*}\triangle DEF\end{align*}, \begin{align*}d=12, E=39^\circ, f=17\end{align*}. Solve the triangle.
- In \begin{align*}\triangle PQR\end{align*}, \begin{align*}P=115^\circ, Q=30^\circ, q=10\end{align*}. Solve the triangle.
- In \begin{align*}\triangle MNL\end{align*}, \begin{align*}m=5, n=9, L=20^\circ\end{align*}. Solve the triangle.
- In \begin{align*}\triangle SEV\end{align*}, \begin{align*}S=50^\circ, E=44^\circ, s=12\end{align*}. Solve the triangle.
- In \begin{align*}\triangle KTS\end{align*}, \begin{align*}k=6, t=15, S=68^\circ\end{align*}. Solve the triangle.
- In \begin{align*}\triangle WRS\end{align*}, \begin{align*}w=3, r=5, s=6\end{align*}. Solve the triangle.
- In \begin{align*}\triangle DLP\end{align*}, \begin{align*}D=52^\circ, L=110^\circ, p=8\end{align*}. Solve the triangle.
- In \begin{align*}\triangle XYZ\end{align*}, \begin{align*}x=10, y=12, z=9\end{align*}. Solve the triangle.
- In \begin{align*}\triangle AMF\end{align*}, \begin{align*}A=99^\circ, m=15, f=16\end{align*}. Solve the triangle.
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Color | Highlighted Text | Notes | |
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Term | Definition |
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law of cosines | The law of cosines is a rule relating the sides of a triangle to the cosine of one of its angles. The law of cosines states that , where is the angle across from side . |
law of sines | The law of sines is a rule applied to triangles stating that the ratio of the sine of an angle to the side opposite that angle is equal to the ratio of the sine of another angle in the triangle to the side opposite that angle. |
Image Attributions
In this Concept we'll use the Pythagorean Theorem, trigonometry functions, the Law of Sines, and the Law of Cosines to solve various triangles. Our focus will be on understanding when it is appropriate to use each method as well as how to apply the methods above in real-world and applied problems.