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# The Law of Cosines Study Guide (page 2)

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Updated on Oct 2, 2011

## Finding Angles with the Law of Cosines

When we know the lengths of all the sides of a triangle, the Law of Cosines can find the measure of any angle. For example, let us find the measures of angles x, y, and z in the triangle pictured in Figure 18.19.

To find the measure of angle x, we use the Law of Cosines with A and B the adjacent edges and C the opposite edge. That is, A = 4, B = 10, C = 7, and θ = x. Thus,

C2 = A2 + B2 – 2AB · cos(θ)
72 = 42 + 102 – 2(4)(10) · cos(x)
49 – 116 = – 80 · cos(x)
x = cos-1(0.8375) ≈ 33.1°

To find the measure of angle y, we use A = 4, B = 7, C = 10, and θ = y. The only really important thing is to make C be the side opposite the angle we are trying to measure.

C2 = A2 + B2 – 2AB · cos(θ)
102 = 42 + 72 – 2(4)(7) · cos(y)
100 = 65 – 56 · cos(y)
y = cos–1(–0.625) ≈ 128.7°

The numbers that come out of the inverse cosine function range from 0° to 180°. This means that we do not have to deal with acute and obtuse angles separately, as we will do in Lesson 19. The acute angle x is estimated at 33.1° and the obtuse angle y is estimated at 128.7° by the same process.

We could use the Law of Cosines to find the measure of angle z, but it would be much easier to use the fact that the three angles of a triangle add up to 180°. Thus,

z ≈ 180° – 33.1° – 128.7° = 18.2°

#### Example

Find the measure of the angle x illustrated in Figure 18.20.

We use the Law of Cosines to find θ = x with adjacent edges A = 9 and B = 10, and opposite edge C = 11.

C2 = A2 + B2 – 2AB · cos(θ)
112 = 92 + 102 – 2(9)( 10) · cos(x)
121 = 181 – 180 · cos(x)
cos(x) =
x = cos–1 ≈ 70.5°

## An Ambiguous Situation

If we know the lengths of two sides of a triangle and the angle between them, then we can use the Law of Cosines to find the length of the third side. If the angle we know is not between the two sides, then there will be ambiguity. For example, look at the two triangles in Figure 18.33.

This situation was discussed at the end of the last lesson. The Law of Sines can be used to find the angles in the lower-right corners of the triangles. However, this process will result in two different angle measures: one acute and one obtuse. Similarly, the Law of Cosines can find the measure the sides labeled x, but it will result in two different numbers. Incidentally, this ambiguity relates to the fact that there is no Angle-Side-Side theorem in geometry, even though there is a Side-Angle-Side theorem.

If we use A = x, B = 10, C = 5, and θ = 25° (remember that the side labeled C must be opposite the given angle), then the Law of Cosines states:

C2 = A2 + B2 – 2AB · cos(θ)
52 = x2 + 102 – 2x(l0) · cos(25)
25 = x2 + 100 – (l8.13)x
x2 – (l8.13)x + 75 = 0

This is a quadratic equation that is not easily factored. Fortunately, it can be evaluated with the quadratic formula.

### Tip

Any quadratic equation ax2 + bx + c = 0 can be solved by the quadratic formula:

To solve x2 – (l8.13)x + 75 = 0, we use the quadratic formula with a = 1, b = –18.13, and c = 75. Thus,

or

The first answer is the x for the triangle on the right of Figure 18.33, and the second is for the triangle on the left.

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