CK-12 Trigonometry

by CK-12 Foundation

  Using the Law of Sines

  In triangle at the right, we know two sides and a non-included angle. Remember that the Law of Sines states: . Since we know , , and , we can use the Law of Sines to find . However, since this is the SSA case, we have to watch out for the Ambiguous case. Since , we could be faced with either Case 1, Case 2, or Case 3 above.

  Since no angle exists with a sine greater than , there is no solution to this problem.

  We also could have compared a and beforehand to see how many solutions there were to this triangle.

  since which tells us there are no solutions.

  In triangle , and . Find .

  Again in this case, and we know two sides and a non-included angle. By comparing a and , we find that:

  since we know that there will be two solutions to this problem.

  There are two angles less than with a sine of , however. We found the first one, , by using the inverse sine function. To find the second one, we will subtract from .

  To check to make sure is a solution, we will use the Triangle Sum Theorem to find the third angle. Remember that all three angles must add up to .

  This problem yields two solutions. Either angle is or .

  We will now refer back to the Real-World Application at the beginning of the section.

  In this problem, we again have the SSA angle case. In order to find the distance from the boat to the lighthouse (a) we will first need to find the measure of angle . In order to find angle , we must first use the Law of Sines to find angle . Since , this situation will yield exactly one answer for the measure of angle .

  First, we will find angle .

  There is another angle less than with a sine of . That angle would be . However, if we add to our other angle of , we exceed , which means is not a solution. Due to the fact that , we already knew that there was only one solution to this problem.

  Now that we know the measure of angle , we can find the measure of angle .

  We can now use the Law of Sines to find side a.

  Answer: The boat is approximately away from lighthouse .

  Using the Law of Cosines

  Real –World Application: In a game of pool, a player must the eight ball into the bottom left pocket of the table. Currently, the eight ball is away from the bottom left pocket. However, due to the position of the cue ball, she must bank the shot off of the right side bumper. If the eight ball is away from the spot on the bumper she needs to hit and forms a angle with the pocket and the spot on the bumper, at what angle does the ball need to leave the bumper?

  In the scenario above, we have the SAS case, which means that we need to use the Law of Cosines to begin solving this problem. The Law of Cosines will allow us to find the distance from the spot on the bumper to the pocket . Once we know , we can use the Law of Sines to find the angle . We will begin by finding .

  The distance from the spot on the bumper to the pocket is . We can now use this distance and the Law of Sines to find angle . We could use the Law of Cosines again, since we now know all three sides of the triangle, but it is more time consuming and requires more computation.

  To find the measure of angle , we will use the Law of Sines. Since we are finding an angle, we are faced with the SSA case, which means we could have no solution, one solution, or two solutions. However, since , we know that this problem will yield only one solution.

  Answer: The ball must leave the bumper at a angle.

  Applications and Tools

  In the previous example, we looked at how we can use the Law of Sines and the Law of Cosines together to solve a problem involving the SSA case. In this section, we will look at situations where we can use not only the Law of Sines and the Law of Cosines, but also the Theorem of Pythagoras and trigonometric ratios. We will also look at another real-world application involving the SSA case.

  Example:

  In triangle at the right, is the altitude of the triangle. If and , find the measure of angle .

  In order to find the measure of angle , we must first know the measure of side . Once we know side , we can use the Law of Sines to solve for angle . To find side , we can use the Theorem of Pythagoras since is a right triangle.

  Now that we know that side is , we can use the Law of Sines to find angle . Since , however, and we have the SSA case, we must watch out for multiple or no solutions. By compare and , we find that and . Since we know that there will be two solutions.

  One possible measure for angle is . To find the other possible measure for angle , we will subtract from .

  Answer: is either or

  Real-World Application:

  Three scientists are out setting up equipment to gather data on a local mountain. Person is away from Person , who is away from Person . Person is away from the mountain. The mountains forms a angle with Person and Person , while Person forms a angle with Person and Person . Find the angle formed by Person with Person and the mountain.

  In the triangle formed by the three people, we know two sides and the included angle (SAS). We can use the Law of Cosines to find the remaining side of this triangle, which we will call . Once we know , we will two sides and the non-included angle (SSA) in the triangle formed by Person , Person , and the mountain. We will then be able to use the Law of Sines to calculate the angle formed by Person with Person and the mountain, which we will refer to as .

  To find :

  Now that we know , we can use the Law of Sines to find . Since this is the SSA case, we need to check to see if we will have no solution, one solution, or two solutions. Since , we know that we will have only one solution to this problem.

  Answer: Person forms an angle of with Person and the mountain.

  Points to Consider

  Why is there only one possible solution to the SSA case if ?

  Explain why yields two possible solutions to a triangle.

  If we have a SSA angle case with two possible solutions, how can we check both solutions to make sure they are correct?

  Lesson Summary

  The SSA case is called the Ambiguous case because two sides and a non-included angle do not necessarily form a unique triangle.

  If side is less than side in the SSA, we could have no solution, one solution, or two solutions. If side is equal to or greater than side , we will have only one triangle.

  If , we can check to see how many solutions a triangle will have by comparing with . If we will have two solutions. If we will have only one solution. If we will have no solution.

  There are many real-world situations where we may be faced with the SSA case in a triangle. We already looked at a few in the example above. We will explore some more scenarios in the review questions.

  Review Questions

  Using the table below, determine how many solutions there would be to each problem based on the given information and by calculating and comparing it with a. Sketch an approximate diagram for each problem in the box labeled “diagram.” If a problem has no solution or two solutions, provide an explanation of why.

  Given , , or Diagram Number of solutions Explanation for or no solutions

  a.

  b.

  c.

  d.

  Using the information in the table above, find all possible measures of angle if any exist.

  Prove using the Law of Sines:

  Give the measure of a non-included angle and the lengths of two sides so that two triangles exist. Explain why two triangles exist for the measures you came up with.

  If and , find the values of so that: There is no solution

  There is one solution

  There are two solutions

  In the figure below, and Find and .

  In the figure below, and Find the following:

  Radio detection sensors for tracking animals have been placed at three different points in a wildlife preserve. The distance between Sensor and Sensor is The distance between Sensor and Sensor is . The angle formed by Sensor with Sensors and is . If the range of Sensor is , will it
be able to detect all movement from its location to the location of Sensor ?

  In problem above, a fourth sensor is placed in the wildlife preserve. Sensor forms a angle with Sensors and , and Sensor forms a angle with Sensors and . How far away is Sensor from Sensors and ?

  Two cell phone companies have towers along Route . Company A’s tower is from one point on Route and from another point. This tower’s signal forms a angle. Company B’s tower is from one point of Route and from another. Company B’s signal forms a angle with the road at the point that is from the tower. For how many miles would a person driving along Route have service with Company A? Company B? For how many miles is there an overlap in coverage?

  Review Answers

  , solutions

  , no solution

  , one solution

  , two solutions

  or

  no solution

  or

  Student answers will vary. Student should mention using in their explanation.

  .

  Yes, it will be able to detect all motion between its location and the location of Sensor .

  Sensor is from Sensor and from Sensor .

  The driver would have service with Company A for and with Company B for . There is of overlap in coverage.

  Supplemental Links

  http://www.algebralab.org/studyaids/studyaid.aspx?file=Trigonometry_LawSines2.xml

  Vocabulary

  Ambiguous case

  A situation that occurs when applying the Law of Sines in an oblique triangle when two sides and a non-included angle are known. The ambiguous case can yield no solution, one solution, or two solutions.

  General Solutions of Triangles

  Learning Objectives

  A student will be able to:

  Use the Theorem of Pythagoras, trigonometry functions, the Law of Sines, and the Law of Cosines to solve various triangles.

  Use combinations of the above methods to solve triangles.

  Understand when it is appropriate to use each method.

  Apply the methods above in real-world and applied problems.

  Introduction

  Real-World Application:

  A cruise ship is based at Island , but makes trips to Island and Island during the day. If the distance from Island to Island is , from Island to is , and Island to is , what heading (angle) must the captain:

  a. Leave Island

  b. Leave Island

  c. Leave Island

  Remember that when using a compass, is due North and is due South which means we must convert our angle measures from the traditional and axis measures.

  In this example, we must calculate all of the angles in the triangle, thereby solving the triangle.

  We will refer back to this application later on.

  In the previous sections we have discussed a number of methods for finding a missing side or angle in a triangle. Previously, we only knew how to do this in right triangles, but now we know how to find missing sides and angles in oblique triangles as well. By combining all of the methods we’ve learned up until this point, it is possible for us to find all missing sides and angles in any triangle we are given.

  In this section, we will review the methods we’ve learned for finding missing angles and triangles and we will combine these methods to solve a number of triangles. In addition, we will look at real-world and application problems that require us to solve different triangles.

  Summary of Triangle Techniques

  Below is a chart summarizing the triangle techniques that we have learned up to this point. This chart describes the type of triangle (either right or oblique), the given information, the appropriate technique to use, and what we can find using each technique.

  Type of Triangle: Given Information: Technique: What we can find:

  Right Two sides Pythagorean Theorem Third side

  Right One angle and one side Trigonometric ratios Either of the other two sides

  Right Two sides Trigonometric ratios Either of the other two angles

  Oblique angles and a non-included side (AAS) Law of Sines The other non-included side

  Oblique angles and the included side (ASA) Law of Sines Either of the non-included sides

  Oblique sides and the angle opposite one of those sides (SSA) – Ambiguous case Law of Sines The angle opposite the other side (can yield no, one, or two solutions)

  Oblique sides and the included angle (SAS) Law of Cosines The third side

  Oblique sides Law of Cosines Any of the three angles

  Using the Law of Cosines

  It is possible for us to completely solve a triangle using the Law of Cosines. In order to do this, we will need to apply the Law of Cosines multiple times to find all of the sides and/or angles we are missing.

  Example 1:

  In triangle

  Solve the triangle.

  Since we are given all three sides in the triangle, we can use the Law of Cosines. Before we can solve the triangle, it is important to know what information we are missing. In this case, we do not know any of the angles, so we are solving for angle , angle , and angle . We will begin by finding .

  The measure of . Now, we can find by again using the Law of Cosines.

  The measure of . We can quickly find by using the Triangle Sum Theorem.

  Answer: , and

  Example 2:

  In triangle and .

  Solve the triangle.

  In this triangle, we have the SAS case because we know two sides and the included angle. This means that we can use the Law of Cosines to solve the triangle. In order to solve this triangle, we need to find side , , and . First, we will need to find side using the Law of Cosines.

  Now that we know , we know all three sides of the triangle. This means that we can use the Law of Cosines to find either angle or angle . We will find angle first. Law of Cosines

  To find angle , we need only to use the Triangle Sum Theorem.

  Answer: , and .

  Real-World Application: A control tower is receiving signals from two microchips implanted in wild tigers. Microchip is from the control tower and microchip is from the control tower. If the control tower forms a angle with both microchips, how far apart are the two tigers? What angle does microchip1 form with the tower and microchip ? What angle does microchip form with the tower and microchip ?

  Part 1: First, we will find the distance between microchip and microchip , which will tell us how far apart the two tigers are. We will call this distance . Since we know two sides and the included angle, we can use the Law of Cosines to find .

  Answer: The two tigers are apart.

  Part 2: Now that we know the third side of the triangle, we can use the Law of Cosines to find either of the other two angles. We will find the angle formed by microchip with the tower and microchip . We will refer to this as angle .

  Answer: The angle formed by microchip with the tower and microchip is .

  Part 3: Now that we know two of the three angles, we can use the Triangle Sum Theorem to find the other angle – the angle formed by microchip with microchip and the tower.

  Answer: The angle formed by microchip with the tower and microchip is

  Using the Law of Sines

  It is also possible for us to completely solve a triangle using the Law of Sines if we begin with the ASA case, the AAS case, or the SSA case. We must remember that when given the SSA case, it is possible that we may encounter the Ambiguous case.

  Example 3:

  In triangle and . Solve the triangle.

  This is an example of the ASA case, which means that we can use the Law of Sines to solve the triangle. In order to use the Law of Sines, we must first know angle , which we can find using the Triangle Sum Theorem.

  Now that we know angle , we can use the Law of Sines to find either side or side . Let’s begin by finding side .

  We can use the same process to find side .

  Answer: and

  We will now refer back to the application at the beginnin
g of the section.

  In order to find all three angles in the triangle, we must use the Law of Cosines because we are dealing with the SSS case. Once we find one angle using the Law of Cosines, we can use the Law of Sines to find a second angle. Then, we can use the Triangle Sum Theorem to find the third angle.

  We could use the Law of Cosines to find all of the angles, but this process is time consuming and requires a lot of computation. Therefore, we will use the Law of Cosines only once in solving this problem.

  When using the Law of Sines after the Law of Cosines to find angles, we have to be aware of the Ambiguous SSA case. In order to avoid the Ambiguous case, we should start by finding the largest angle, which is across from the largest side. The largest angle has the greatest chance of being obtuse. So, if we find that angle first, we won’t have to worry about the Ambiguous case.

  We will begin by finding angle since it is the largest angle.

  Now that we know , we can find either or . We will find first since it is the second largest angle.