The Law of COSINES

The Law of COSINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides:

The Law of COSINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing angles:

SAS - 2 sides and their included angle SSS Use Law of COSINES when ... you have 3 dimensions of a triangle and you need to find the other 3 dimensions . They cannot be just ANY 3 dimensions though, or you won’t have enough information to solve the Law of Cosines equations. Use the Law of Cosines if you are given: SAS - 2 sides and their included angle SSS

Example 1: Given SAS Find all the missing dimensions of triangle ABC, given that angle B = 98°, side a = 13 and side c = 20. Use the Law of Cosines equation that uses a, c and B to find side b: B 98° C a = 13 A c = 20 b

Example 1: Given SAS Now that we know B and b, we can use the Law of Sines to find one of the missing angles: B 98° C a = 13 A c = 20 b = 25.3 Solution: b = 25.3, C = 51.5°, A = 30.5°

Example 2: Given SAS Find all the missing dimensions of triangle, ABC, given that angle A = 39°, side b = 20 and side c = 15. Use the Law of Cosines equation that uses b, c and A to find side a: 39° A b = 20 c = 15 B C a

Example 2: Given SAS Use the Law of Sines to find one of the missing angles: 39° A b = 20 c = 15 B C a = 12.6 Important: Notice that we used the Law of Sine equation to find angle C rather than angle B. The Law of Sine equation will never produce an obtuse angle. If we had used the Law of Sine equation to find angle B we would have gotten 87.5°, which is not correct, it is the reference angle for the correct answer, 92.5°. If an angle might be obtuse, never use the Law of Sine equation to find it.

Example 3: Given SSS Find all the missing dimensions of triangle, ABC, given that side a = 30, side b = 20 and side c = 15. We can use any of the Law of Cosine equations, filling in a, b & c and solving for one angle. Once we have an angle, we can either use another Law of Cosine equation to find another angle, or use the Law of Sines to find another angle. A C B a = 30 c = 15 b = 20

Example 3: Given SSS Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it. For this reason, we will use the Law of Cosines to find the largest angle first (in case it happens to be obtuse). Angle A is largest because side a is largest: A C B a = 30 c = 15 b = 20

Example 3: Given SSS A B C Solution: A = 117.3° B = 36.3° C = 26.4° Use Law of Sines to find angle B or C (its safe because they cannot be obtuse): Solution: A = 117.3° B = 36.3° C = 26.4°

The Law of Cosines SAS SSS When given one of these dimension combinations, use the Law of Cosines to find one missing dimension and then use Law of Sines to find the rest. Important: The Law of Sines will never produce an obtuse angle. If an angle might be obtuse, never use the Law of Sines to find it.

Review Use Law of SINES when given ... AAS ASA SSA (the ambiguous case)

Heron’s Formula

Heron’s Area Formula The Law of Cosines can be used to establish the following formula for the area of a triangle. This formula is called Heron’s Area Formula after the Greek mathematician Heron (c. 100 B.C.).

Area of a Triangle Law of Cosines Case - SSS B C c a b h SSS – Given all three sides Heron’s formula:

Example 5 – Using Heron’s Area Formula Find the area of a triangle having sides of lengths a = 43 meters, b = 53 meters, and c = 72 meters. Solution: Because s = (a + b + c)/2 = 168/2 = 84, Heron’s Area Formula yields  1131.89 square meters.

USING HERON’S FORMULA TO FIND AN AREA (SSS) The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of Earth.)

USING HERON’S FORMULA TO FIND AN AREA (SSS) (continued) The semiperimeter s is Using Heron’s formula, the area  is

Four possible cases can occur when solving an oblique triangle. When to use the Law of Sines and the Law of Cosines Four possible cases can occur when solving an oblique triangle.

Try these Given the triangle with three sides of 6, 8, 10 find the area Given the triangle with three sides of 12, 15, 21 find the area

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