Law of Sines The Law of Sines is

Law of Sines The Law of Sines is used to find missing information in an oblique triangle (a triangle that does not contain a right angle). General Formula To use the Law of Sines, you need to have 4 pieces of information. Of those 4 pieces, 3 will be known and 1 will be unknown. C a b A c B Maybe we should look at an example with actual numbers.

Law of Sines Example In triangle ABC, a = 12 inches, angle A = 37 o, and angle B = 24 o. What is the length of side b, to the nearest tenth of an inch? First, let’s draw a diagram. C b A 37 o 12 24 o B

Another Law of Sines Example In triangle ABC, b = 100 meters, angle A = 59 o, and angle B = 74 o. What is the length of side c, to the nearest tenth of a meter? First, let’s draw a diagram. Since we are not given the measurement of the angle opposite the side that we need to find, we must first find the measurement of angle C. Now we can use the Law of Sines to find side c. C 100 A 47 o 59 o 74 o c B

More Law of Sines In triangle ABC, a = 23 inches, angle A = 41 o, and b = 17 inches. What is the measurement of angle B, to the nearest whole inch? That was easy C 17 A 41 o 23 B When you’re finding the measurement of an angle, you use the 2 nd key on your calculator.

The Double Triangle Problem The angle of elevation from a ship at point A to the top of a lighthouse at point B is 43 o. When the ship reaches point C, 300 meters closer to the lighthouse, the angle of elevation is 56 o. Find, to the nearest meter, the height of the lighthouse. B 13 o 5 9. 90 A 56 o 43 o 300 3 C D

Another Double Triangle Problem The angle of depression from the top of a lighthouse at point A to a ship at point B is 59 o. When the ship reaches point C, 210 feet closer to the lighthouse, the angle of depression is 71 o. Find, to the nearest foot, the height of the lighthouse. A 71 o 59 o 12 o 8 7 5. 86 B 59 o 210 71 o C D That was easy

Law of Cosines Once again, to use the Law of Cosines, you need to have 4 pieces of information. Of those 4 pieces, 3 will be known and 1 will be unknown. Depending on what the given information is, the formula to use the Law of Cosines would be: C or a b A or That looks really confusing. c B Let’s look at an example.

Law of Cosines Example In triangle ABC, a = 6, b = 10, and angle C = 120 o. What is the length of side c? C 10 A 120 o c 6 B

Another Law of Cosines Example Boris (B) and Chester (C) have computer factories that are 132 miles apart. They both ship computer parts to Aquafina (A). Aquafina is 72 miles from Boris, and 84 miles from Chester. If points A, B, and C are located on a map, find to the nearest tenth of a degree, the measure of the largest angle in the triangle formed. A 84 72 B 132 C

Area of a Triangle The Area of a Triangle can be expressed in terms of the lengths of any two sides and the sine of the included angle. Sounds like there’s a formula coming. You must use two sides and the angle included between those two sides. So, depending on what the given information is, the area of this triangle can be represented as: C a b A c That looks confusing. B or or

Area Example Let’s throw some numbers into the mix and see if we can actually find the area of a triangle. If angle A = 150 o, b = 8 centimeters, and c = 10 centimeters, find the area of triangle ABC. A 10 150 o First, let’s draw a diagram. 8 C B That was easy

Another Area Example Sometimes if we know the area of a triangle, we can use the area to help find other pieces of the triangle. The area of an isosceles triangle is 49 in 2, and the measure of the congruent sides is 14 inches each. Find the measurement of the angle included between the two congruent sides. That was easy A May I push the easy button? 14 14 K = 49 in 2 B C

Yet Another Area Example The area of triangle ABC is 50 in 2. If a = 9 inches, and the measurement of angle C is 72 o. Find, to the nearest tenth of an inch, the measurement of side b. A b K = 50 in 2 B 72 o 9 This is just way too easy! C