4.8/5.5/5.6 Right Triangle Trig Applications Law of Sines & Law of Cosines

Transcription

1 Objective: 4.8/5.5/5.6 Right Triangle Trig Applications Law of Sines & Law of Cosines Apply right triangle trigonometry. Solve triangles using the Law of Sines and the Law of Cosines. WARMUP Find the missing angles and sides of the triangle. Assume that angle B is a right angle. I. Angles of Elevation and Depression An angle of elevation can best be described as the angle through which the eye moves as it moves from looking horizontally (straight ahead) up to something above. An angle of depression is the angle through which the eye moves as it moves from looking horizontally (straight ahead) down to something below.

2 How To Make an Inclinometer!

3 How To USE an Inclinometer!

4 You, Your Inclinometer, and Trig What More Is There Really? For #1-3 below, you must use your knowledge of right triangle trigonometry to find the requested information. Feel free to measure things using your inclinometer and/or a tape measure. 1) Pick an object and mark your location. Measure the distance from where you re standing to the object. Then note the angle of inclination from the horizontal to the top of the object. Using these measurements, find the height of your object. My object is I am standing feet from it The angle of inclination I measured is So..the height of my object 2) Pick an object and mark your location. Measure the distance from where you re standing to the object. Then measure the height of the object. Using these measurements, find the angle of inclination to your object. My object is I am standing feet from it The height of my object is So..the angle of inclination to my object is Now, check that with your inclinometer. What angle did your inclinometer get?

5 3) Time to climb! Find a location in the football bleachers. One of you should SAFELY go to a higher part of the bleachers. The other should stay on level ground below. Measure how high your partner is off of the ground on which you stand. Measure the horizontal distance between you and a direct line from your partner to the ground. Using these measurements, the higher partner should find the angle of depression and the lower partner should find the angle of elevation to your object. The higher of us is feet off the ground The lower of us is feet away from a direct spot right below the higher person So..the angle of elevation to my partner is So..the angle of depression to my partner is Now, check that with your inclinometer. What angle did your inclinometer get? II. Law of Sines sin A sin B sinc a b c This law states that the ratio of the sine of an angle to the length of its opposite side is the same for all three angles of any triangle. It will be helpful in finding missing sides or angles of a NON-RIGHT TRIANGLE! Example 1 Solve ABC given that A 36, B 48, a 8

6 Example 2 Solve ABC given that a 7, b 6, A Example 3 Smokey Don Bramley, who works as a fire ranger in the summer, sights a fire at Ranger Station A in the direction of 32 east of north. Fire Marshall Cahill spots the same fire from Ranger Station B, which stands 10 miles to the east. Cahill sights the fire on a line 48 west of north. Who is closer to the fire? How much closer?

7 III. Example 4 Law of Cosines a b c 2bc cos A b a c 2ac cos B c a b 2ab cosc Solve ABC given that a 11, b 5, C 20. Example 5 Solve ABC given that a 9, b 7, c 5.

8 Example 6 In the Winter Wipeout version of Temple Run, two snowmobilers begin their journey by heading in opposite directions. Indiana Jones rides 2 miles one way while Tom Glenn rides 3 miles in another direction. At what angle must they have diverged at in the beginning if they are now 3.5 miles apart?

9 Applying the Laws Google Earth Style! In this activity, we will use the Law of Sines and the Law of Cosines to measure various distances around the MHS campus. The one thing we know is that the football field (including the end zones is 120 yards long (what is that in feet?). For each new point, you need to draw a picture of the triangle on this sheet, fully labeled, and show all of your work. You can find angles using a protractor, but you may NOT use a ruler to find distance! You may use previous questions to answer later questions. You should use the unit feet for the entire activity! 1) Following the diagram below, solve the resulting triangle formed by working across any two parking spaces in the lot behind the school. The dashed lines in the diagram indicate the triangle we are working with. You should be able to accurately measure all the sides and then use the appropriate law to find the missing angles. SCHOOL 2) Using your Google map: (a) Find the distance of the side of the football field closest to the bleachers. (b) Find the distance from the northeastern most corner of the football field to the pitcher s mound on the varsity baseball field. (c) Find the distance from the pitcher s mound back to the northwestern most corner of the football field (AKA where you started in part a) (d) Solve the resulting triangle and don t forget a diagram!

10 3) Using your Google map: (a) Find the distance from the northwestern most corner of the football field to home plate of the JV softball field. (b) Find the distance from the rear entrance in center hall of the high school to home plate of the JV softball field. (c) Find the distance from the rear entrance in center hall of the high school to the northwestern most corner of the football field. Bonus: Imagine that a zip line is to be tied from the top of the roof covering the rear entrance to the high school and extends to the 50 yard line of the football field. How much line is required?