SOH CAH TOA only works in right triangles! Law of Cosines If you do not have a right triangle you use Law of Sines or Law of cosines. SOH CAH TOA only works in right triangles!

Law of cosines: c2 = a2+b2 – 2abcos C Other two sides C Opposite side and angle b a A B c

Law of cosines: c2 = a2+b2 – 2abcos C Finding a Missing Side Length 45 m x 36o 52 m Step 1: Identify the opposite side/angle Step 2: Plug into law of cosines formula Step 3: Solve using your calculator Step 4: Last step is to take the square root To use law of cosines to find a side you must have SAS

Example 1 Step 1: identify the opposite side/angle Step 2: plug into law of cosines formula Step 3: solve using your calc Step 4: last step is to take the square root 45 x 36o 52

I must see this step on your homework for credit! Example 2 70o 22 31 I must see this step on your homework for credit! x

Example 3 Two 11-cm radii of a circle form a central angle measuring 115o. What is the length of the chord connecting the two radii? Draw and label a picture x2 = 112 +112 – 2(11)(11)cos(115) x2= 344.27 x = 18.6 cm 11 115 11

Finding Missing Angle Measures 250 175 E Q 225 Note: When given SSS, it is always best to solve for either your biggest or smallest angle first so you have a reference point when finding the other two angles!

Last step is inverse (you are looking for an angle) Example 4: Find Q You must solve this in steps! Start with the side across from the angle you are looking for D CLT 250 175 -81250 -81250 Undo +/- -78750 E Q -78750 Undo ×/÷ 225 .2381=cosQ Last step is inverse (you are looking for an angle) Use cos(.2381) Q = 76.2o

C 70 B A 62 Example 5: Find all of the missing angles 51 1. Find the largest angle first (remember, the largest angle is across from the largest side) C 70 59° 51 CLT Undo +/- 76o 45° B A 62 Undo ×/÷ 2. Find the smallest using law of sines Use cos(.2443) Find the last angle by subtracting the other two from 180 C =180 – (76+45)= 59° Cross multiply!!

Example 6 The diagonals of a parallelogram are 60 in and 70in and intersect at an angle measuring 64o. Find the length of the shorter side of the parallelogram. Round to the nearest tenth place. Draw a picture 70 35 60 x 64o 30

Law of Cosines Worksheet Homework: Law of Cosines Worksheet