Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8)

Transcription

1 Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8) Standards: Geom 19.0, Geom 20.0, Trig 7.0, Trig 8.0, Trig 12.0 Segerstrom High School -- Math Analysis Honors Name: Period: Thinkbinder Study Group: Reminders: Practice Problems (PQ & PT) are completed in spiral bound notebook only. All pages in spiral notebook should be labeled accordingly: Unit Concept - (title of assignment) Examples: Unit O Concept 1 Practice Quiz Unit O Concept 1-4 Practice Test Website with all video links and resources: kirchmathanalysis.blogspot.com Edmodo Group Codes for class communication: Success is not the key to happiness. Happiness is the key to success. If you love what you are doing, you will be successful. Herman Cain Concept Mandatory # What we will be learning Practice 1 evaluate angles in degrees or radians on a calculator Practice quiz 1 Page 275 #51-68; page 285 # finding angles on a calculator using inverse trig functions Practice quiz 2 Page 327 #1-9; page 328 # solving basic trig equations with non-exact answers Practice quiz 3 4 finding all six trig ratios given point on terminal side Practice quiz 4 Page 294 # finding all six trig ratios when given one trig ratio (must draw right triangle) 6 using SOHCAHTOA to find missing pieces of right triangles (given 2 sides, given angle and side) 7 using the triangle Practice quiz 7 Practice quiz 5 Page 284 # using the triangle Practice quiz 8 9 solving basic right triangle word problems Practice quiz 9 10 solving angle of elevation and depression word problems Practice quiz 10 Optional Extra practice from textbook Practice quiz 6 Page 284 #1-8; page 337 #1-10 Page 286 #69-76 Page 266 #59-64; Page 286 #77-84; page 337 #17-32 This Unit is all about our favorite type of triangles: RIGHT TRIANGLES! There are lot of special relationships that only exist when a triangle has a RIGHT ANGLE, so these triangles are super cool! We will start with basic use of the trig functions on our calculator, both regular trig functions and INVERSE trig functions. Hopefully the word INVERSE means a lot more to you now than it used to! A trig function and an inverse trig function will undo each other and will have the same relationships that all other inverses have numerically, algebraically, and graphically. We will review our basic ratios of SOHCAHTOA that we learned in geometry, and see how those ratios apply to the unit circle ratios we looked at in Unit J. Putting all of our knowledge together, including the PYTHAGOREAN THEOREM, we will solve right triangles meaning to find ALL MISSING SIDES and ALL MISSING ANGLES. Two special triangles exist that we will also solve, but we will be able to solve them using patterns instead of by using calculations. Lastly, we will explore applications of right triangles, starting with the use of the PYTHAOGREAN THEOREM, and then using angles of elevation and depression. It is most important to remember that angles of elevation always go UP from a horizontal and angles of depression always go DOWN from a horizontal.

2 evaluate angles in degrees or radians on a calculator To use the reciprocal trig functions, we must do this: CSC (cosecant) = SEC (secant) = COT (cotangent) = We have TWO modes that our calculator can be in when dealing with trigonometry: DEGREES or RADIANS. **You must make sure your calculator is in the mode you want it to be!!!

3 finding angles on a calculator using inverse trig functions x+3 = 5 sin(x)=.7845 How would you get rid of the 3? Why does that work? How do you get rid of the cube root? Why does that work? How do you get rid of the Why does that work? How do you get rid of the sin? Why does that work? sin(x) = : use sin -1 (x) to solve cos(x) = : use cos -1 (x) to solve tan(x) = : use tan -1 (x) to solve If your calculator is in DEGREE MODE, the answer they give you will be in DEGREES! If your calculator is in RADIAN MODE, the answer they give you will be in RADIANS! *My tip: Leave calculator in degree mode, and just CONVERT answers to radians at the end as necessary* WHAT IS??? It is the Greek letter for THETA, and it is a variable (just like x ) that is commonly used in trigonometry! These problems ask for the answers for one revolution of the Unit Circle. However, your calculator only gives you ONE answer. You need to do two things: 1. Make sure that answer is within one revolution of the Unit Circle example: sin(x) = The calculator will tell you the answer is o. That is not within one revolution. Find a coterminal angle by adding 360 o to find that o is the real angle the calculator is talking about. 2. Find the second answer using reference angles o has a reference angle of o (found by using ). Because the problem was sin(x) = -.876, our two answers will lie in the two quadrants where sine is negative. These quadrants are III and IV o is in Quadrant IV, so we just have to find the angle in Quadrant III that has a reference angle of o. We do this by taking 180 o and adding o to it, giving us o as our second answer. Quadrants 1 st answer 2 nd answer Quadrants 1 st answer 2 nd answer

4 solving basic trig equations with non-exact answers Let s take this one step further and solve trig equation where we have to do a little algebra first nothing from concept 2 changes! Simplified equation Quadrants Answer #1 Answer #2

5 finding all six trig ratios given point on terminal side y represents the opposite side Using the Pythagorean theorem, we know that x represents the adjacent side r represents the hypotenuse Sine: sin Cosecant: csc Cosine: cos Secant: sec Tangent: tan Cotangent: cot Example: Find all six trig functions given that a point on the terminal side of the angle is (-3,-4) In this problem, x=-3 and y=-4 sin = -4/5 csc = -5/4 Using Pythagorean Theorem, r=5 Because the terminal side is in Quadrant III, only tangent and cotangent should have positive values. cos = -3/5 sec = -5/3 tan = 4/3 cot = 3/4

6 x= y= r= Quadrant: ; cos is + / - x= y= r= Quadrant: ; sin is + / - x= y= r= Quadrant: ; csc is + / - x= y= r= Quadrant: ; tan is + / - x= y= r= Quadrant: ; cos is + / - x= y= r= Quadrant: ; cot is + / - x= y= r= Quadrant: ; sin is + / - x= y= r= Quadrant: ; sec is + / - x= y= r= Quadrant: ; sin is + / - For extra practice, find ALL SIX trig functions for these problems!

7 finding all six trig ratios when given one trig ratio (must draw right triangle) 1. Take the trig function you are given and draw a Quadrant I right triangle labeled appropriately. 2. Find the missing side of the triangle. 3. Use the given information to find the requested trig ratio. 4. **Find all trig ratios for great practice! sec =hyp/adj or r/x y 2 = 25 2 y 2 =49 y=7 cos = x/r = 24/25 sin = y/r = 7/25 tan = y/x = 7/24 csc = r/y = 25/7 cot = x/y = 24/7

8 using SOHCAHTOA to find missing pieces of right triangles (given 2 sides, given angle and side) To SOLVE a right triangle, it means to find ALL MISSING SIDES and ALL MISSING ANGLES. Just for fun (to the tune of We Will Rock You : Pythagorean Theorem goes with right triangles Hypotenuse is across from the right angles Legs are a and b Hypotenuse c Plug the numbers in and see how easy it can be! Example: b=11, a= 7 Angles are labeled with CAPITAL letters Sides are labeled with LOWERCASE letters Sides and angles correspond with each other because they are ACROSS from each other. C is the right angle always! Use the Pythagorean Theorem and Inverse Trig Functions to solve! Note: *Never use a rounded value in the middle of a problem unless there are no other options left! We must find c, A, and B. Let s start with c = c 2 c = 13.0 (always check to make sure it doesn t simplify; GIVE BOTH EXACT RADICAL AND APPROXIMATION) tana=7/11 tan -1 (7/11) = 32.5 o tanb=11/7 tan -1 (11/7) = 57.5 o

9

10 using the triangle AND using the triangle See answers to all SSS examples above + 10 more examples to try of each type in the turquoise box of the blog. Label each side according to the pattern n, n, n Solve for n first, and then find all the missing sides.

11 Label each side according to the pattern n, 2n, n Solve for n first, and then find all the missing sides.

12 Combining the Two Directions: Find the missing values. Access PQ7-8 worksheet on the blog for triangle pictures.

13 solving basic right triangle word problems One of the most common formulas used with right triangles is the Pythagorean Theorem!

14 solving angle of elevation and depression word problems Angles are sometimes measured in DEGREES and MINUTES (and sometimes SECONDS!), instead of giving a decimal approximation. This is similar to how latitute and longitude values are labeled. Example: 36 o 4 is 36 degrees, 4 minutes.

15 3a) A man at ground level measures the angle of elevation to the top of a building to be 67 o. If, at this point, he is 15 feet away from the building, what is the height of the building? 3b) The same man now stands atop a building. He measures the angle of elevation to the building across the street to be 27 o and the angle of depression (to the base of the building across the street) to be 31 o. If the two buildings are 50 feet apart, how tall is the taller building? 5) From a point 120 feet from the base of a church, the angles of elevation of the top of the building and the top of a cross are 38 o and 43 o, respectively. a) Find the height of the cross. (The ground is flat). b) Find the height of the building (not including the cross) 6) A man is just about to ski down a steep mountain. He estimates the angle of depression from where he is now to the flag at the bottom of the course to be 24 o. He knows that he is 800 feet higher than the base of the course. How long is the path that he will ski? (Round to the nearest foot).

16

17 Unit O Practice Quiz 1 There is always the possibility that there is an error in the answer key (WE ALL MAKE MISTAKES SOMETIMES!) If you believe you have found an error, please submit the Submit an Error form that you ll find on the right hand side of the kirchmathanalysis.blogspot.com site. Mrs. Kirch will double-check and verify. THANK YOU FOR YOUR HARD WORK AND ATTENTION TO DETAIL! Unit O Practice Quiz 2 Unit O Practice Quiz 3 Unit O Practice Quiz 4 Unit O Practice Quiz 5 Unit O Practice Quiz 6 Unit O Practice Quiz 7-8 See KEY AT END OF DOCUMENT on blog Unit O Practice Quiz in miles Unit O Practice Quiz degrees meters degrees ,278.3 feet yards 6..2 miles degrees 3. 4 feet 4. NO (does not satisfy Pyth Thm) 8. No, 6.3 feet meters meters

18 Unit O Practice Test Use the same directions from the PQ problems for each concept to solve these in order to prepare for the test. Answer key is posted online only at kirchmathanalysis.blogspot.com Concept 1 Concept 2 Concept 3 Concept 4 Concept 5 Concept 6 Concept 7-8 Concept 9 Finish the rest of the PQ problems 1. To get from point A to point B you must avoid walking through a pond. To avoid the pond, you must walk 34 meters south and 41 meters east. To the nearest meter, how many meters would be saved if it were possible to walk through the pond? 2. A baseball diamond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between first base and third base? 3. A suitcase measures 24 inches long and 18 inches high. What is the diagonal length of the suitcase to the nearest tenth of a foot? 4. In a computer catalog, a computer monitor is listed as being 19 inches. This distance is the diagonal distance across the screen. If the screen measures 10 inches in height, what is the actual width of the screen to the nearest inch?

19 Concept 10 Unit Circle values/angles (Unit N Concept 8-9) Video answer key: Review Solving 3-variable systems (Unit J Concept 3)