Law of Cosines

Let me paint you a picture. Two students, Sam and Monica are comparing how long each of them takes to cycle to school from their homes. They have deduced that Sam's home is 4 miles away from school while Monica's is 3 miles away. They have also found out that the angle formed by the two distances from their homes to the school is 43o. Let us sketch this out. We find that a triangle is constructed as shown below.

Law of Cosines Law of Cosines

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    Real-world example 1, Aishah Amri - StudySmarter Originals

    Now, from this information, is there some sort of method by which we can determine the distance between Sam and Monica's house? Fortunately, you are in luck! There is indeed a formula that can help us solve this problem. We can use the Law of Cosines. In this topic, we shall be introduced to the Law of Cosines and its application to solving triangles.

    Law of Cosines

    The Law of Cosines is defined by the rule

    \[a^2=b^2+c^2-2ab\cos(A)\]

    Before we begin, let us ease ourselves into this topic by recalling the Cosine Ratio and Pythagoras' Theorem. That way we can bridge the two concepts and derive the Law of Cosines as mentioned above.

    Recap: The Cosine Ratio and Pythagoras' Theorem

    Let us first recall the Pythagoras Theorem for a given right-angle triangle with an angle θ.

    Right-angle triangle, Aishah Amri - StudySmarter Originals

    Thus is given by,

    \[c^2=a^2+b^2\].

    Furthermore, recall that the cosine of an angle is found by dividing the adjacent by the hypotenuse. This is known as the Cosine Ratio

    \[\cos(\theta=\frac{adjacent}{hypotenuse}=\frac{b}{c}\].

    Derivation of the Law of Cosines

    Now that we have set the ideas above, let us use them as a building block in establishing the Law of Cosines. Consider the triangle ABC below.

    Law of Cosines, Aishah Amri - StudySmarter Originals

    The triangle ABC is split into two triangles: ABD and BCD. Both these triangles are right triangles divided by the perpendicular leg BD of measure h. The side b is also divided into two sections AD = x and CD = b – x. Furthermore, the hypotenuse of triangle ABD is AB = c while the hypotenuse of triangle BCD is BC = a.

    Let us analyse the relationship between the lengths of a, b, c and the angle A. By Pythagoras Theorem, we can solve triangle DBC by

    \[a^2=(b-x)^2+h^2\]

    Expanding (b - x)2, we obtain

    \[a^2=b^2-2bx+x^2+h^2\]

    Similarly, we can do this for triangle ADB as

    \[c^2=x^2+h^2\]

    Substituting this expression for x2 + h2 into the previous equation, we obtain

    \[a^2=b^2-2bx+c^2\]

    Notice that we can further write x in the form of a Cosine Ratio as

    \[\cos(A)=\frac{x}{c}\Rightarrow x=c\cdot \cos(A)\]

    Replacing this expression for x, we obtain

    \[a^2=b^2-2bc\cdot \cos(A)+c^2\]

    Applying the Commutative Property and rearranging this, we obtain the Law of Cosines

    \[a^2=b^2 +c^2 -2bc\cdot \cos(A)\]

    as we have stated at the beginning of this section.

    Variations of the Law of Cosines

    The Law of Cosines has two other variations from the one given above. By the Law of Cosines, we see that the formula gives us the measure of one unknown side of a triangle. However, a triangle has three sides and three angles. In some cases, we may need to apply this concept on any of the other sides should we require to find their measures. Let us consider the triangle ABC below.

    Variations of the Law of Cosines, Aishah Amri - StudySmarter Originals

    Here, the value of a, b and c represent the length of each side of this triangle. Angles A, B and C describe the opposite angles for each of these sides respectively. There are three forms of the Law of Cosines for this triangle.

    \begin{align} a^2&=b^2+c^2-2bc\cos(A)\\b^2&=a^2+c^2-2ac\cos(B)\\ c^2&=a^2+b^2-2ab\cos(A)\end{align}

    The forms above are suitable for finding the length of an unknown side given the measures of two sides and their included angle. However, if we are given the values of three sides, we may need to perform some lengthy algebra to determine the unknown angle. To simplify such calculations, we can rearrange the expressions above so that we get an explicit formula for the unknown angle.

    \begin{align} \cos(A)&=\frac{b^2+c^2-a^2}{2bc}\\\cos(B)&=\frac{a^2+c^2-b^2}{2ac} \\\cos(C)&=\frac{b^2+b^2-c^2}{2ab}\end{align}

    Application of the Law of Cosines

    In this section, we shall observe several worked examples that apply the Law of Cosines. We can apply the Law of Cosines for any triangle given the measures of two cases:

    1. The value of two sides and their included angle

    2. The value of three sides

    Solving a Triangle Given Two Sides and their Included Angle

    Solve a triangle ABC given b = 17, c = 16 and A = 83o.

    Solution

    Let us make a sketch of this triangle.

    Example 1, Aishah Amri - StudySmarter Originals

    We begin by using the Law of Cosines to find the length of a.

    \begin{align}a^2&=b^2+c^2-2ac\cdot \cos(A)\\a^2&=17^2+16^2-2\cdot 17\cdot 16\cdot\cos(83)\\a^2&=545-544\cdot\cos(83)\\\Rightarrow a&=\sqrt{535-544\cdot(\cos(83)}\\a&\approx 21.88\end{align}

    Thus, a is approximately 21.88 units. From here, we shall use the Law of Sines to find angle C.

    \begin{align} \frac{\sin(A)}{a}&=\frac{\sin(C)}{c}\\\Rightarrow \sin(C)&=\frac{16\cdot \sin(83)}{21.88}\\\Rightarrow C&\approx 46.54^\circ\end{align}

    Thus, C is approximately 46.54o. Finally, we can find angle B by

    \begin{align} A+B+C&=180^\circ\\B&=180^\circ -A-C\approx 50.46^\circ\end{align}

    Thus, B is approximately 50.46o.

    Solving a Triangle Given Three Sides

    Solve the triangle below given a = 14, b = 11 and c = 5.

    Solution

    We begin by using the Law of Cosines to find angle A.

    \begin{align} \cos(A)&=\frac{b^2+c^2-a^2}{2bc}\\\cos(A)&=\frac{11^2+5^2-14^2}{2\cdot 11 \cdot 5}\\\cos(A) &=-\frac{5}{11}\\\Rightarrow A&\approx 117.04^\circ\end{align}

    Thus, A is approximately 117.04o. From here, we shall use the Law of Sines to find angle B.

    \begin{align}\frac{\sin(A)}{a}&=\frac{\sin(B)}{b}\\\Rightarrow \sin(B)&=\frac{11\cdot \sin(117.04)}{14}\\B&\approx 44.41^\circ\end{align}

    Thus, B is approximately 44.41o. Finally, we can find angle C by

    \[A+B+C=180^\circ\Rightarrow C=180^\circ-A-B\approx 18.55^\circ\]

    Thus, C is approximately 18.55o.

    Real-World Example Involving Law of Cosines

    Randy is training for a marathon. He starts his journey by running 9 miles in one direction. Then, he turns and runs another 11 miles. The two legs of his run form an angle of 79o. How far is Randy from his starting point at the end of the 11-mile leg of his run?

    Solution

    Let us begin by sketching the outline of this problem. The variable d represents the distance from Randy's starting point and the end of his 11-mile leg run.

    Real-world example 2, Aishah Amri - StudySmarter Originals

    Here, we are given the lengths of two sides and their included angle. Thus, we can use the Law of Cosines to find d.

    \begin{align} d^2&=9^2+11^2-2\cdot 9\cdot 11\cdot \cos(79)\\d^2&=202-198\cdot (79)\\d&=\sqrt{202-198\cdot cos(79)}\\d&\approx 12.81\ miles\end{align}

    Therefore, d is approximately 12.81 miles.

    Find the step angle X made by the hindfoot of a person whose pace averages 27 inches and stride averages 32 inches.

    Solution

    Let us begin by sketching the outline of this problem. The green dot represents each step made by the hindfoot of a person.

    Real-world example 3, Aishah Amri - StudySmarter Originals

    Here, we are given the lengths of three sides. Thus, we can use the Law of Cosines to find the step angle, X.

    \begin{align} \cos (X)&=\frac{32^2+27^2-27^2}{2\cdot 32\cdot 27}\\ \cos(X)&=\frac{1024}{1728}\\\cos(X) &=\frac{16}{27}\\ X&\approx 53.66^\circ\end{align}

    Therefore, X is approximately 53.66o.

    Solving Oblique Triangles

    An oblique triangle is one that has no right angle. Below are some examples of oblique triangles.

    Oblique triangles, Aishah Amri - StudySmarter Originals

    To solve such triangles, we need the length of at least one side and the value of any other two components of a given triangle. If the triangle has a solution, then we must choose whether to begin our solution by applying the Law of Sines or the Law of Cosines. The table below helps us decide on such situations.

    Given ComponentsMethod to Begin With
    Two angles and any sideLaw of Sines

    Two sides and an angle opposite one of them

    Law of Sines
    Two sides and their included angleLaw of Cosines
    Three sidesLaw of Cosines

    Law of Cosines - Key takeaways

    • The Law of Cosines states that

    \begin{align} a^2&=b^2+c^2-2bc\cos(A)\\b^2&=a^2+c^2-2ac\cos(B)\\ c^2&=a^2+b^2-2ab\cos(A)\end{align}

    • Similarly, this can be written in the forms

    \begin{align} \cos(A)&=\frac{b^2+c^2-a^2}{2bc}\\\cos(B)&=\frac{a^2+c^2-b^2}{2ac} \\\cos(C)&=\frac{b^2+b^2-c^2}{2ab}\end{align}

    • We can use the Law of Cosines to solve triangles given
      • The value of two sides and their included angle

      • The value of three sides

    Frequently Asked Questions about Law of Cosines

    What is the law of cosines?

    Given a triangle ABC, the law of cosines states that a2 = b2 + c- 2bc Cos A

    What is law of cosines used for? 

    The law of cosines is used to find an unknown side of a triangle given the value of two sides and their included angle or to find an unknown angle given three sides of a triangle

    What is the formula when solving the law of cosines?

    The formula for the law of cosines is a2 = b2 + c- 2bc Cos A

    What are real world examples of law of cosines? 

    The law of cosines can be used to measure distance and angles of elevation 

    How do you proof the law of cosines? 

    You can prove the law of cosines by Pythagoras Theorem

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