Objective: To apply the Law of Cosines for finding the length of a missing side of a triangle. Lesson 18 Law of Cosines

When solving an oblique triangle, if two sides AND the angle BETWEEN then is know, the Law of Cosines allows us to find the third side. If all 3 sides are known we can find an angle. The equations are as follows: The Law of Cosines

The formula for the Law of Cosines makes use of three sides and the angle opposite one of those sides. We can use the Law of Cosines: a. if we know two sides and the included angle, or b. if we know all three sides of a triangle. General Strategies for Using the Law of Cosines

87.0° c   From the model, we need to determine c, , and . We start by applying the law of cosines. Two sides and one angles are known. SAS

To solve for the missing side in this model, we use the form: In this form,  is the angle between a and b, and c is the side opposite . 87.0° c   a b

Using the relationship c 2 = a 2 + b 2 – 2ab cos  We get c 2 = – 2(15.0)(17.0)cos 87.0° = – 510(0.0523) = Soc = 22.08

Now, since we know the measure of one angle and the length of the side opposite it, we can use the Law of Sines to complete the problem. and This gives and Note that due to round-off error, the angles do not add up to exactly 180°.

Three sides are known. SSS In this figure, we need to find the three angles, , , and .

To solve a triangle when all three sides are known we must first find one angle using the Law of Cosines. We must isolate and solve for the cosine of the angle we are seeking, then use the inverse cosine to find the angle.

We do this by rewriting the Law of Cosines equation to the following form: In this form, the square being subtracted is the square of the side opposite the angle we are looking for Angle to look for Side to square and subtract

We start by finding cos 

From the equation we get and

° Our triangle now looks like this: Again, since we have the measure for both a side and the angle opposite it, we can use the Law of Sines to complete the solution of this triangle.

° Completing the solution we get the following: and

Solving these two equations we get the following: and Again, because of round-off error, the angles do not add up to exactly 180 .

Find x using the Law of Cosines x² = 10² + 14² −2(10)(14)cos(44°) x² = 296 −280cos(44) x² = 94.6 x = 9.73 Practice

Solving SAS Triangles Example. Problem: If a = 5, c = 9, and β = 25 °, find b, α and γ

Solving SSS Triangles Example. Problem: If a = 7, b = 4, and c = 8, find α, β andγ

If the angle that we know is not between the two sides, then there will be an ambiguous case. (There is no Angle-Side-Side theorem) The Ambiguous Case 10 5 x 25 o 10 5 x 25 o

10 5 x 25 o 10 5 x 25 o We need the quadratic formula to solve this (can’t easily factor)

Find two possible lengths for the side x Practice o x