17.+Solve+bearing+and+orienteering+problems+with+law+of+sines+and+law+of+cosines.

//Solve bearing and orienteering problems wit////h law of sines and law of cosines:// Serina Adler Honors Pre-Calculus  Period 1 The reason for why I chose this section involves the importance of this section. The two trigonometry functions, law of cosines and law of sines, are of crucial importance in trigonometry based on the following reasons: 1. //law of sines// is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are given and 2. //law of cosines// can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known. Furthermore, bearings are significant as they are used a lot to find the direction of an object. In other words, a bearing will tell you the direction from one point to another relative point. Thus this section is very important when it comes to understanding trigonometry. Not only did I choose this section because of its importance, but I also chose it because it was my favorite section. It was my favorite section because I enjoy doing algebraic problems as opposed to graphing, or other math techniques. I found this section definitely has its difficulties, but overall I found this section easy to comprehend. **Law of Cosines: **

The law of cosines relates to the length of the sides of a plane triangle to the cosines of one of its angle. //a// and //b// stand for the opposite side lengths of //c//, which are given to you. //C// stands for the angle between the two side lengths given. //c// stands for the side length, opposite of //C//.

The formula can be written in three different ways. This is depending on what sides and angles you are given and/or if you change which sides of the triangle play the role of a, b, and c.  **Law of Sines:** The law of sines is an equation relating the lengths of the sides of a triangle to the sines of its angle.  //a, b,// and //c// are the lengths of the sides of the triangles. //A, B,// and //C// are the opposite angles <span style="display: block; font-family: Georgia,serif; text-align: center;">(as shown in the picture above) <span style="background-color: #ffffff; display: block; font-family: Georgia,serif; text-align: center;">The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known. <span style="display: block; font-family: Georgia,serif; text-align: center;">Click here if you want to know more about law of sines!  <span style="font-family: Georgia,serif; font-size: 160%;">**What is a be****a****ring?**  <span style="font-family: Georgia,serif;">There are many different types of bearings.   <span style="font-family: Georgia,serif;">A bearing is an angle that is measured in relation to the fixed horizontal reference plane of true north, that is using the direction toward the geographic north pole as a reference. Instead of standard position, the angle is measured in degrees in a clockwise direction from the north line (this is also known as a __true bearing__).   <span style="font-family: Georgia,serif;">Bearings can be used for many different things. In aircraft navigation, a bearing is the actual (corrected) compass direction of the forward course of the aircraft. In land navigation, a bearing is the angle between a line connecting two points and a north-south line. To put it in simpler terms, a bearing will tell you the direction from one point to another relative point. <span style="font-family: Georgia,serif; font-size: small; line-height: 24px;">**Bearings are significant because true bearings are often used instead of compass bearings, like at airports. At an airport, the numbers represent four quadrants in which the difference between the numbers equal 9. What one may not realize though, is that the numbers on the runway are missing a zero, meaning the actual difference is 90. This would mean the total is 360, which is relative the amount of degrees on a compass.** <span style="font-family: Georgia,serif; font-size: 110%; line-height: 24px;">**For example:**  <span style="font-family: Georgia,serif; font-size: small; line-height: 24px;">16 - 7 = 9 ** → **<span style="font-family: Georgia,serif; font-size: small; line-height: 24px;">160 -70 = 90  <span style="font-family: Georgia,serif; line-height: 24px;">25 - 16 = 9 ** → **<span style="font-family: Georgia,serif; font-size: small; line-height: 24px;">250 - 160 = 90  <span style="font-family: Georgia,serif; line-height: 24px;">34 - 25 = 9 ** → **<span style="font-family: Georgia,serif; line-height: 24px;">340 - 250 = 90 **<span style="font-family: Georgia,serif; font-size: 160%;">How do you solve a bearing? ** <span style="font-family: Georgia,serif;">Instead of starting on the right horizontal line, like you would from standard position, 0 degrees starts at the top vertical line of the quadrants -North. From 0 degrees, you would go in a clockwise direction. <span style="font-family: Georgia,serif;"> __<span style="font-family: Georgia,serif; font-size: 120%;">For Example __<span style="font-family: Georgia,serif; font-size: 120%;">:

<span style="font-family: Georgia,serif; text-align: right;">The angle/bearing of point P is 48 degrees.

<span style="display: block; font-family: Georgia,serif; text-align: left;">The angle/bearing of the plane is 30 degrees. <span style="display: block; font-family: Georgia,serif; text-align: left;">The angle/bearing for point A is 30 degrees. <span style="display: block; font-family: Georgia,serif; text-align: left;">The angle/bearing for point C is 110 degrees. <span style="display: block; font-family: Georgia,serif; text-align: left;">The angle/bearing for point D is 260 degrees. <span style="display: block; font-family: Georgia,serif; text-align: left;">The angle/bearing for point B is 300 degrees. //<span style="font-family: Georgia,serif; font-size: 150%;">Example Problems: // <span style="font-family: Georgia,serif;">Solve the bearing for the following problems. <span style="font-family: Georgia,serif;"> <span style="font-family: Georgia,serif;">What is the bearing for point P? <span style="font-family: Georgia,serif;">a) 60 degrees <span style="font-family: Georgia,serif;">b) 210 degrees <span style="font-family: Georgia,serif;">c) 120 degrees <span style="font-family: Georgia,serif;">d) 240 degrees

<span style="font-family: Georgia,serif;">What is the bearing for point P? <span style="font-family: Georgia,serif;">a) 140 degrees  <span style="font-family: Georgia,serif;">b) 40 degrees  <span style="font-family: Georgia,serif;">c) -50 degrees  <span style="font-family: Georgia,serif;">d) 310 degrees <span style="font-family: Georgia,serif;">What is the bearing for point P? <span style="font-family: Georgia,serif;">a) 160 degrees <span style="font-family: Georgia,serif;">b) 290 degrees <span style="font-family: Georgia,serif;">c) 70 degrees <span style="font-family: Georgia,serif;">d) 20 degrees

**<span style="font-family: Georgia,serif; font-size: 160%;">Applying what we know to solve bearing and orienteering problems with Law of Sines and Law of Cosines: ** //<span style="font-family: Georgia,serif; font-size: 160%;">Example Problems: // <span style="font-family: Georgia,serif; font-size: 110%; line-height: 31px;">1. <span style="font-family: Georgia,serif; font-size: 160%;"> <span style="font-family: Georgia,serif; text-align: center;">In the example, the bearing of the plane is 270° and the bearing of the wind is 225°. Redrawing the figure as a triangle using the tail-tip rule, the length (ground speed of the plane) and bearing of the resultant can be calculated <span style="background-color: #ffffff; color: #333333; font-family: Georgia,serif; font-size: 14px;">. <span style="background-color: #ffffff; color: #000000; font-family: Georgia,serif; font-size: 60%;">(http://www.cliffsnotes.com/study_guide/Vector-Operations.topicArticleId-11658,articleId-11621.html)

  <span style="display: block; font-family: georgia,serif; text-align: center;">First, use the law of cosines to find the magnitude of the resultant. 





<span style="display: block; font-family: georgia,serif; text-align: center;">Then, use the law of sines to find the bearing. 





<span style="display: block; font-family: georgia,serif; text-align: center;">The bearing, β, is therefore 270° − 4.64°, or approximately 265.4°.

<span style="display: block; font-family: georgia,serif; font-size: 14px; text-align: left;">2. 

<span style="font-family: Georgia,serif;">A plane flies at 300 miles per hour. There is a wind blowing out of the southeast at 86 miles per hour with a bearing of 320°. At what bearing must the plane head in order to have a true bearing (relative to the ground) of 14°? What will be the plane's groundspeed ? <span style="background-color: #ffffff; display: block; font-family: Georgia,serif; font-size: 8px; text-align: center;">(http://www.cliffsnotes.com/study_guide/Vector-Operations.topicArticleId-11658,articleId-11621.html)

  <span style="display: block; font-family: georgia,serif; text-align: center;">Use the law of sines to calculate the bearing and the groundspeed. Because these alternate interior angles are congruent, the 54° angle is the sum of the 14° angle and the 40° angle.



<span style="display: block; font-family: georgia,serif; text-align: center;">Therefore, the bearing of the plane should be 14° + 13.4° = 27.4°. The groundspeed of the plane is 342.3 miles per hour.

**<span style="font-family: Georgia,serif; font-size: 160%;">//Example Problem:// ** <span style="font-family: Georgia,serif;">A frisbee is thrown at 15mph at a bearing of 80 degrees. There is a wind from the southeast going 10mph at a bearing of -45 degrees. Use law of cosines and law of sines to determine the magnitude and angle of the resultant of the frisbee. **<span style="font-family: Georgia,serif; font-size: 170%;">? ? ? ? ? ? ? ? ? ** **<span style="font-family: Georgia,serif; font-size: 170%;">[|Click here for instructional video!] ** <span style="font-family: Georgia,serif;">__ Works Cited: __ <span style="display: block; font-family: georgia,serif; text-align: center;"> http://www.mathsteacher.com.au/year7/ch08_angles/07_bear/bearing.htm <span style="display: block; font-family: georgia,serif; text-align: center;"> http://mathsclass.net/comments/bearing-and-airports <span style="display: block; font-family: georgia,serif; text-align: center;"> http://www.cliffsnotes.com/study_guide/Vector-Operations.topicArticleId-11658,articleId-11621.html <span style="display: block; font-family: georgia,serif; text-align: center;"> http://en.wikipedia.org/wiki/Law_of_sines <span style="display: block; font-family: georgia,serif; text-align: center;"> http://en.wikipedia.org/wiki/Law_of_cosines