## Law of Sines and Cosines, and Areas of Triangles – She Loves Math

Students will practice solve problems involving the ambiguous case of the law of sines to solve a variety of problems including word problems. X Advertisement. Example Questions. A triangle has two sides with lengths of 20 and The measure of the angle opposite the side with a length of 15 is 35°. Find all the possible measures of the angle. Feb 25,  · Learn how to determine if a given SSA triangle has 1, 2 or no possible triangles. Given two adjacent side lengths and an angle opposite one of . The process for solving Law of Sines: Ambiguous Case Triangles is really simple because all you have to do is grab some FRUIT! Fruit? It’s my acronym for how to solve Triangles involving the Ambiguous Case, and it’s really easy. So, how do you find “FRUIT” and solve .

## The Ambiguous Case of the Law of Sines

Note that the second set of three trig functions are just the reciprocals of the first three; this makes it a little easier! The only problem is that sometimes, with the SSA case, depending on what we know about the other sides and angles of the triangle, the triangle could actually have two different shapes one acute and one obtuse. Note that we usually depict angles in capital lettersand the sides directly across from them in the same letter, but in lower case ; see left-most column for Law of Sines:.

Use Law of Cosines when you have these parts of a Triangle in a row:. Note: When using the Law of Cosines to solve the whole triangle all angles and sidesparticularly in the case of an obtuse triangle, you have to either finish solving the whole triangle using Law of Cosines which is typically more difficultor use the Law of Sines starting with the next smallest angle the angle across from the smallest side first.

This is because of another case of ambiguous triangles. I tend to round the angle measurements to a tenth of a degree, and the side measurements two decimal places hundredths. Solve for a :. Solve for A :. We have an SSA case that happens not to be the ambiguous case. So, we have:. Solve ambiguous case applications B :. Again, solving the triangle means finding all the missing parts, both sides and angles. Solving for b and cwe get:. Solving for a and bwe get:, ambiguous case applications.

Now we can complete the table:. When we have the Side-Side-Angle in a row case SSAwe could have onetwoor no triangles formed, and we have to do extra work to determine which situation we have. We can then solve for two different triangles the given two sides and one angle for the two triangles will be the same.

This happens when the height of the triangle equals the paired side the side across from the known angle, ambiguous case applications. Solve for all possible triangles with the given conditions:. Now we can get Angle Aand solve for side a in both cases.

Note how the original values Bambiguous case applications, band c stay the same in both cases:. Again, note that we usually depict angles in capital lettersand the sides directly across from them in the same letter, but in lower case ; see right-most column for Law of Cosines:. Note that angle A is called the included angle between sides b ambiguous case applications csince it is located between the two sides. No triangle can be formed with these side lengths.

Caution : When using the Law of Cosines to solve the whole triangle all angles and sidesparticularly in the case of an obtuse triangle, you have ambiguous case applications either finish solving the whole triangle using Law of Cosines which ambiguous case applications typically more difficultor use the Law of Sines starting with the next smallest angle the angle across from the smallest side first, ambiguous case applications.

This is because of another example of ambiguous cases with triangles. We can do this fairly easily using a graphing calculator ; in fact the calculator can actually ambiguous case applications us how many triangles we will get! Now that we have the two answers for cwe can use the Law of Sines to solve for the two solutions for angle C :, ambiguous case applications.

Now that we know trig, we get the area of a triangle without having to know the altitude if we know two sides, and the angle inside the two sides the Side-Angle-Side or SAS caseor three sides of the triangle Side-Side-Side, or SSS case.

A little bit more complicated, but not too bad! For example, draw the angles as close to the correct angle measurements and sides in the proportion of the numbers they give you. The distance from Ali to the plane is roughly 1 By definition, a parallelogram is a quadrilateral four-sided figure with straight sides that has opposite parallel sides, and it turns out that opposites sides are equal. Parallel means never crossing, like railroad tracks.

These are called Same Side Interior angles, ambiguous case applications. And each time a boat or ship changes course, you have to draw another line to the north to map its new bearing.

Also, remember from Geometry that Alternate Interior Angles are congruent when a transversal cuts parallel lines. It then travels ambiguous case applications mph for 2 hours. Find the distance the ship is from its original position and also its bearing from the ambiguous case applications position. Similarly, we get 20 miles by multiplying 10 mph by 2 hours.

The distance the ship is from its original position is As shown in the figure below, Joa is standing feet from her friend Rachel. What is the distance from Emily to Rachel? Probably the most difficult part is to drawing a picture of the problem:. The distance from Emily to Rachel is about 61 feet. Adding the two areas together, we see that the total area of the piece of land is Understand these problems, and practice, practice, practice!

Use the Right Triangle Button on the MathType keyboard to enter a problem, and then click on Submit the arrow to the right of the problem to solve the problem. You can also click on the 3 dots in the upper right hand corner to drill down for example problems. You can even get math worksheets. There is even a Mathway App for your mobile device. Skip to content. Remember that the sin cos, and so on of an angle is just a number! No triangle exists! Note how the original values Bband c stay the same in both cases: Triangle 1: Angles:.

Triangle 1: Angles:. Triangle ambiguous case applications Angles:, ambiguous case applications. Ambiguous case applications this Area Formula ambiguous case applications you have these parts of a Triangle in a row:. As shown in the picture below, Ali ambiguous case applications Brynn are standing yards apart. How far from the plane is Ali? What is the measure of angle BCD? Three dogs are sitting in a kitchen and waiting to get their dog food.

Dog A is 4, ambiguous case applications. How far is Dog C from the dog food? Jill, a surveyor, needs to approximate the area of a piece of land, as shown below.

She walks the perimeter of the land and measures the side distances and one angle, as shown below. What is the area of the piece of land? Note: Separate the piece of land into two triangles:.

### Law of Sines or Sine Rule (solutions, examples, videos) The process for solving Law of Sines: Ambiguous Case Triangles is really simple because all you have to do is grab some FRUIT! Fruit? It’s my acronym for how to solve Triangles involving the Ambiguous Case, and it’s really easy. So, how do you find “FRUIT” and solve . Precalculus Help» Trigonometric Applications» Law of Sines» The Ambiguous Case (SSA, ASS) Example Question #1: The Ambiguous Case (Ssa, Ass) Given and, . Feb 25,  · Learn how to determine if a given SSA triangle has 1, 2 or no possible triangles. Given two adjacent side lengths and an angle opposite one of .