## Law of Cosines

### Form 1:

$a 2 = b 2 + c 2 − 2 b c cos ⁡ A$

### Form 2:

$b 2 = a 2 + c 2 − 2 a c cos ⁡ B$

### Form 3:

$c 2 = a 2 + b 2 − 2 a b cos ⁡ C$

### Example 1: Find length of segment $A B ¯$ ### step 1: Determine which form of the Law of Cosines to use based on given information Given Information Angles Sides A = unknown a = 6.7 B = unknown b = 5.0 C = 65° c = answer Because the question is asking us to find $A B ¯$ or side c, we will use form 3 of the equation.

$c 2 = a 2 + b 2 − 2 a b cos ⁡ C$

### step 2: rewrite equation in terms of c instead of c by taking the square root of both sides of the equation.2

$c 2 = a 2 + b 2 − 2 a b cos ⁡ C$
$c 2 = a 2 + b 2 − 2 a b cos ⁡ ( C )$
$c = a 2 + b 2 − 2 a b cos ⁡ ( C )$

### step 3: Plug in known values to the equation found in step 2

$c = 6.7 2 + 5.0 2 − 2 (6.7) (5.0) cos ⁡ ( 65° )$

$c = 6.45$

### Example2: Find the measure of angle B ### step 1: Determine which form of the Law of Cosines to use based on given information Given Information Angles Sides A = unknown a = 3.7 B = answer b = 6.5 C = unknown c = 8.0 Because the question is asking us to find angle B, we will use form 2 of the equation.

$b 2 = a 2 + c 2 − 2 a c cos ⁡ B$

### step 2: Rewrite the equation to isolate the answer, B.

$b 2 = a 2 + c 2 − 2 a c cos ⁡ B b 2 − a 2 − c 2 = − 2 a c cos ⁡ B b 2 − a 2 − c 2 − 2 a c = cos ⁡ B$

### To isolate B and get rid of Cos( ) we apply the inverse trig function which is arcos or cos-1 in this case.

$cos − 1 ⁡ ( b 2 − a 2 − c 2 − 2 a c ) = cos − 1 ⁡ ( cos ⁡ B ) cos − 1 ⁡ ( b 2 − a 2 − c 2 − 2 a c ) = B$

### step 3: Plug in known values to equation found in step 2

$cos − 1 ⁡ ( 6.5 2 − 3.7 2 − 8.0 2 − 2 (3.7) (8.0) ) = B$

$B = 52.23 °$