Example: Two wires help support a high mast. One wire forms a 48° angle with the ground and the other wire forms a 72° angle with the ground. The wires are 20m apart. How far is the pole? The two sets of solutions are: ∠ Q = 84.2°, ∠ R = 39.8°, r = 7.72 cm ∠ Q = 95.8°, ∠ R = 28.2°, r = 5.70 cm The leaning tower of Pisa is inclined 5.5 degrees from the vertical. At a distance of 100 meters from the tower wall, the angle of height from the top is 30.5 degrees. Do you use the law of sins to estimate the height of the leaning tower? The ambiguous case of solving triangles with Sines` law given SSA How can you get two solutions using Sines` law? Example: A group of foresters crossed Denali National Park on their way to Mount McKinley, the highest mountain in North America. From their campsite they can reach Mt. McKinley, and the elevation angle of their campground at the summit is 21°. You know that the slope of the mountain forms a 127° angle to the ground and that the vertical height of Mount McKinley is 20,320 feet. How far is your campsite from the foot of the mountain? If they can travel 2.9 miles in an hour, how long will it take them to get to base? Now we can use the law of sins to determine the elimination of fire from Station A.
These two problems of the law of sins below will show you how to apply the law of sins to solve certain problems in real life. We can also write the sine law or the sinusoidal rule as: We can use the law of sines when solving triangles. To solve a triangle is to find the unknown lengths and angles of the triangle. If we get two sides and a closed angle (SAS) or three sides (SSS), we use the law of cosine to solve the triangle. The law of sines is also known as the sinusoidal rule, sinusoidal law or sinusoidal formula. It applies to all types of triangles: rectangle, pointed or blunt. Example: A person has a kite on a 1750-foot rope at a 75° angle to the ground. An observer notes that the angle formed by the kite and the plane is 102°. How far is the kite from the observer? These lessons with examples, solutions and videos to help high school students apply the law of sin. This video solves the following application with the law of the sine.
The following video shows the proof of the law of sins. How to solve a word problem that uses the law of Sines? The hardest thing here is creating the graph. We show it below. Note that the height is indicated by a green line. Two fire stations are 15 miles apart, with Station A located directly east of Station B. Both stations discovered a fire. The angular fire direction for Station B is N52°E and the angular fire direction for Station A is N36°W. How far is the light from Station A? Related pages Law of cosines Trigonometry lessons More algebra 2 lessons Sines` law states that: In a given triangle, the ratio of the length of one side and the sine of the angle to that side is a constant. Law of Sines – SSA – How to know if there are 0, 1 or 2 solutions How to use the law of sines to determine if there are 0, 1 or 2 solutions. One method to solve a missing length or angle of a triangle is to use the law of sine. The law of sine, unlike the law of cosine, uses proportions to resolve missing lengths. The ratio of the sine of one angle to the opposite side is the same for the three angles of a triangle.
The law of sine works for any triangle, not just right-angled triangles. Sine Law – Ambiguous Case (SSA) – No Solution If two sides and an unclosed angle (SSA) are given in a triangle, this is called an ambiguous case for the sine law. If you specify these values, you have 0, 1 or 2 possible solutions to solve the given triangle. The solution is ∠ Q = 35.8° , ∠ R = 28.2° and r = 4.36 cm ∠ Q = 35.8°, ∠ R = 180° – 116° – 35.8° = 28.2° (1) Determine whether the following measurements result in a triangle, two triangles or no triangles: (15) Two vehicles leave the same place P simultaneously on two different roads. One vehicle travels at an average speed of 60 km/h and the other vehicle at an average speed of 80 km/h. After half an hour, the vehicle reaches destinations A and B. If AB underlies 60◦ at the starting point P, then find AB. Solution Also note that the angle opposite to 15 is missing, so we need to find it.
Here`s what the chart will look like after drawing it. (14) A man starts his morning walk at point A reaches two points B and C and finally returns to A, so that ∠A = 60◦ and ∠B = 45◦, AC = 4 km in triangle ABC. Find out the total distance he walked on his morning walk. Solution (8) A researcher wants to determine the width of a pond from east to west, which is not possible by actual measurement. From point P, the distance to the easternmost point of the pond is 8 km, while the distance to the westernmost point of P is 6 km. If the angle between the two sightlines is 60°, you will find the width of the pond. The solution Q cannot be an obtuse angle, since the sum of the angles in the triangle exceeds 180°. The only valid value for Q is 35.8°. The law of sines can be used to calculate the remaining sides of a triangle when two angles and one side are known (AAS or ASA) or when we get two sides and one unclosed angle (SSA).