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MAT 151: College Algebra with Trigonometry

Section 4.2 The Law of Sines and Law of Cosines

Goal:
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  • T: Be able to solve problems involving the law of sines or law of cosines.
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The Law of Sines and the Law of Cosines are formulas that are used to solve for unknown values in a triangle. While the SOHCAHTOA definitions apply only in right triangles, the law of sines and law of cosines apply in all triangles. Whether you need one or the other (or both) depends on what you know and what you need to know.
Several triangles are shown on a coordinate plane with labeled vertices. On the left, triangle ABC is drawn with sides labeled a, b, and c. To the right, three separate triangles are shown: triangle DEF with a marked angle at D, triangle EFG with a marked angle at E, and triangle HGK with a marked angle at H. Line segments connect the labeled points to form each triangle, and the marked angles are highlighted to indicate angle relationships within the triangles.
Figure 4.2.1.
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We will state the laws for \(\triangle ABC\text{.}\) The Law of Sines states:
\begin{equation*} \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}. \end{equation*}
The Law of Cosines states:
\begin{equation*} c^2=a^2+b^2-2ab\cos C. \end{equation*}
Note that while the Law of Cosines looks a little like the Pythagorean Theorem, and is in fact related to it, when using the law, it does not matter which angle acts as \(C\text{.}\)
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Apply the Law of Sines or Law of Cosines to solve the following problems:
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Problem 4.2.2.

Suppose that in \(\triangle DEF\text{,}\) we know that \(DF=1.84\) cm, \(m \angle DEF=.415\) radians, and \(m \angle FDE =1.303\) radians. Find the lengths of the other two sides of the triangle.
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Problem 4.2.3.

Suppose that in \(\triangle GHK\text{,}\) \(GH=3.93\) cm, \(HK=8.51\) cm, and \(m \angle H = \frac{\pi}{6}\text{.}\) Find the length \(GK\text{.}\)
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Problem 4.2.4.

Two wires are going in opposite directions from the top of a post to the ground. The angle between the two wires is \(85^{\circ}\) degrees . The ends of the wires are 15 feet apart on the ground. One of the angles forms an angle of \(35^{\circ}\) with the ground. Find the lengths of the wires.
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Problem 4.2.5.

Two friends start at the same point and begin walking away from each other. Each person is walking in a straight line, and the angle made by their two directions is \(\frac{5}{9} \pi\text{.}\) After an hour, one person has walked 5 miles and the other has walked 6.5 miles. How far apart are they?
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Problem 4.2.6.

Recall Figure 2.2.12 from a previous section. Fireworks are being shot out of a cannon at an angle of \(\frac{5 \pi}{12}\) radians with the ground. You are standing 200 m behind the cannon, and you observe that the angle of elevation from the ground where you are standing to where the fireworks are exploding is \(\frac{\pi}{4}\) radians.
The diagram shows a person standing on the ground and fireworks exploding in the air. There is a line that connects the person to the fireworks, and the fireworks cannon to the fireworks. The angle between those two lines closest to the fireworks is labeled alpha.
Figure 4.2.7.
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