The Law of SINES

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

Use Law of SINES when ... AAS - 2 angles and 1 adjacent side you have 3 dimensions of any triangle and you need to find the one or more of the other dimensions Use the Law of Sines if you are given: AAS - 2 angles and 1 adjacent side ASA - 2 angles and their included side SSA- 2 sides and a obtuse angle opposite them

#1. You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measure of side b . A C B 70° 80° a = 12 c b

#2, Find c Fill length of side b Find angle C Set up the Law of Sines to find side c: B 80° a = 12 c 70° 30° A C b = 12.6

We now know all the side lengths and angle sizes of the triangle C B 70° 80° a = 12 c = 6.4 b = 12.6 30° Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm Note: We used the given values of A and a in both calculations. Your answer is more accurate if you do not used rounded values in calculations.

We MUST find angle A first because the only side given is side a. #3. You are given a triangle, ABC, with angle C = 115°, angle B = 30° and side a = 30 cm. Find the measures of angle A and sides b and c. A C B 115° 30° a = 30 c b To solve for the missing sides or angles, we must have an angle and opposite side to set up the first equation. We MUST find angle A first because the only side given is side a. The angles in a ∆ total 180°, so angle A = 35°.

#3 continued Set up the Law of Sines to find side b: 30° 35° A C B 115° 30° a = 30 c b 35° Set up the Law of Sines to find side b:

#4. Find side c Set up the Law of Sines to find side c: 30° 35° A C B 115° 30° a = 30 c b = 26.2 35° Set up the Law of Sines to find side c:

Example 2 (solution) Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm 115° 30° a = 30 c = 47.4 b = 26.2 35° Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm Note: Use the Law of Sines whenever you are given 2 angles and one side!

C SOLUTION: 10 12 A 180 B 14.90 = A

Solving an SAS Triangle The Law of Sines is good for ASA - two angles and the included side AAS - two angles and any side SSA - two sides and an opposite angle (being aware of possible ambiguity) Why would the Law of Sines not work for an SAS triangle? 26° 15 12.5 No side opposite from any angle to get the ratio

Deriving the Law of Cosines Write an equation using Pythagorean theorem for shaded triangle. b h a k c - k A B c

Law of Cosines Visualize the Law of Sines and Law of Cosines Geogebra

Law of Cosines Law of Cosines can be used to solve a non-right triangle when you know one angle and two sides Note the pattern

Applying the Cosine Law Now use it to solve the triangle we started with Label sides and angles Side c first C 15 26° 12.5 A B c

Applying the Cosine Law Now calculate the angles use and solve for B C 15 26° 12.5 A B c = 6.65

Applying the Cosine Law The remaining angle determined by subtraction 180 – 93.75 – 26 = 60.25 C 15 26° 12.5 A B c = 6.65

Wing Span The leading edge of each wing of the B-2 Stealth Bomber measures 105.6 feet in length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)? Hint … use the law of cosines! C A

Using the Cosine Law to Find Area Recall that We can use the value for h to determine the area C b h a A B c

Using the Cosine Law to Find Area We can find the area knowing two sides and the included angle Note the pattern C b a A B c

Try It Out Determine the area of these triangles 42.8° 127° 76.3° 17.9 24 12 76.3°

Cost of a Lot An industrial piece of real estate is priced at $4.15 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 324 feet by 516 feet by 412 feet. 324 412 516