# Using the Law of Cosines to Find the Third Side

In which c is the side across from angle C.

Just like the Law of Sines,

**the Law of Cosines works for any triangle**, not just right triangles.

In particular, the Law of Cosines can be used to find the length of the third side of a triangle when you know the length of two sides and the angle in between.

To use the Law of Sines to find a third side:

1. Identify angle C. It is the angle whose measure you know.

2. Identify a and b as the sides that are not across from angle C.

3. Substitute the values into the Law of Cosines.

4. Solve the equation for the missing side.

2. Identify a and b as the sides that are not across from angle C.

3. Substitute the values into the Law of Cosines.

4. Solve the equation for the missing side.

**Examples:**

1. Find the length of the third side:

We must first determine which angle is C. We are told the measure of angle E, so that will be C in our formula. Thus the empty side will be side c and 6 and 7 will be a and b:

Now we will subsitute into the formula:

And solve the equation:

Carefully input this into our calculator:

c ≈ 9.6 in.

2. Find the length of side c if , a = 5 ft. and b = 8 ft.

We must first determine which angle is C. We are told the measure of angle E, so that will be C in our formula. Thus the empty side will be side c and 6 and 7 will be a and b:

Now we will subsitute into the formula:

And solve the equation:

Carefully input this into our calculator:

c ≈ 9.6 in.

2. Find the length of side c if , a = 5 ft. and b = 8 ft.

We could draw a picture, but since everything is clearly labeled, we don't need to. Our known angle is already called angle C and the sides are already labeled as a andb, so we are ready to subsitute:

And solve the equation:

Carefully input this into our calculator:

c ≈ 6.0 ft.

And solve the equation:

Carefully input this into our calculator:

c ≈ 6.0 ft.

**Practice:**Use the Law of Cosines to find the length of the third side for each triangle. Round your answer to the nearest tenth.

**Hint:**Draw a picture if needed.

1. , a = 10 cm. and b = 6 cm.

2. , a = 2 in. and b = 9 in.

3. , a = 4 m. and b = 2 m.

4. , y = 8 mm. and z = 5 mm.

5. , d = 9 ft. and e = 12 ft.

2. , a = 2 in. and b = 9 in.

3. , a = 4 m. and b = 2 m.

4. , y = 8 mm. and z = 5 mm.

5. , d = 9 ft. and e = 12 ft.

**Answers:**1) 11.0 cm 2) 7.2 in. 3) 3.9 m. 4) 6.9 mm. 5) 18.5 ft.

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