Warm-up A farmer has a triangular field where two sides measure 450 yards and 320 yards.  The angle between these two sides measures 80º.  The farmer wishes to use an insecticide that costs $4.50 for every 100 square yards.  What will it cost to use this insecticide on this field?

Law of Sines

Objective Students will be able to…use the Law of Sines to solve triangles and real-world problems.

In the past you’ve used trig to solve right triangles. Here’s the situation… In the past you’ve used trig to solve right triangles. To solve a triangle with no right angle, you need to know the measure of at least one side and any two other parts of the triangle.

Law of Sines Works in the following cases: ASA (Angle-Side-Angle) AAS (Angle-Angle-Side) SSA (Side-Side-Angle) **this could be the ambiguous case

Law of Sines In trigonometry, we can use the Law of Sines to find missing parts of triangles that are not right triangles. Law of Sines: In ABC, sin A = sin B = sin C a b c B A C c b a

Example 1a: Find p. Round to the nearest tenth.

Example 1a: Law of Sines Cross products Divide each side by sin Use a calculator. Answer:

Example 1b: to the nearest degree in , Law of Sines Cross products Divide each side by 7.

Example 1b: Solve for L. Use a calculator. Answer:

Your Turn: a. Find c. b. Find mR to the nearest degree in RST if r = 12, t = 7, and mT = 76. Answer: Answer:

Solving a Triangle The Law of Sines can be used to “solve a triangle,” which means to find the measures of all of the angles and all of the sides of a triangle.

Example 2a: . Round angle measures to the nearest degree and side measures to the nearest tenth. We know the measures of two angles of the triangle. Use the Angle Sum Theorem to find

Example 2a: Angle Sum Theorem Add. Subtract 120 from each side. Since we know and f, use proportions involving

Example 2a: To find d: Law of Sines Substitute. Cross products Divide each side by sin 8°. Use a calculator.

Example 2a: To find e: Law of Sines Substitute. Cross products Divide each side by sin 8°. Use a calculator. Answer:

Example 2b: Round angle measures to the nearest degree and side measures to the nearest tenth. We know the measure of two sides and an angle opposite one of the sides. Law of Sines Cross products

Example 2b: Divide each side by 16. Solve for L. Use a calculator. Angle Sum Theorem Substitute. Add. Subtract 116 from each side.

Example 2b: Law of Sines Cross products Divide each side by sin Use a calculator. Answer:

Your Turn: a. Solve Round angle measures to the nearest degree and side measures to the nearest tenth. b. Round angle measures to the nearest degree and side measures to the nearest tenth. Answer: Answer:

Example 3: A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is Draw a diagram Draw Then find the

Example 3: Since you know the measures of two angles of the triangle, and the length of a side opposite one of the angles you can use the Law of Sines to find the length of the shadow.

Example 3: Law of Sines Cross products Divide each side by sin Use a calculator. Answer: The length of the shadow is about 75.9 feet.

Your Turn: A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water? Answer: About 42 feet of the fishing line that is cast out is above the surface of the water.