6.1 Angles in Standard Position In Quadrant 1.notebook May 24, 2015

Transcription

1 Unit 6 Trignmetry 6.1 Angles in Standard Psitin in Quadrant 1 1

2 6.1 Trignmetry: Angles in Standard Psitin Outcmes 6.1, 6.2 & 6.6 Determine & sketch the angle in standard psitin given a pint P(x, y) n the terminal arm f the angle & determine the reference angle fr its angle in standard psitin. Fr a pint n the crdinate plane, when an angle θ is between 0 and 360, the angle is measured cunterclckwise frm the psitive x-axis, the initial arm. This angle is said t be in standard. psitin The line jining the pint and the rigin is called the terminal arm. The pint P is the terminal. pint 2

3 Trignmetry Key Terms: The Cartesian Plane: has fur quadrants that run cunterclckwise starting at the tp right quadrant which is psitive (x, y). Initial Arm: the psitive x axis; the starting pint f the rtatin. II y Terminal Arm: the stpping pint f the rtatin. x C terminal Angle: Tw r mre angles in standard psitin that have the same terminal arm. Reference Angle: The acute angle between the terminal arm and the x axis. General Slutin: All pssible slutins as a functin f n. 3

4 Angles in Standard Psitin in Quadrant I Outcme 6.1 Sketch an angle in standard psitin, given the measure f the angle. Using the diagram belw, write an expressin fr: - r in terms f x and y r P(x, y) - the value f θ in terms f x and y - the x-crdinate f P in terms f r and θ. - the y-crdinate f P in terms f r and θ. 4

5 Outcmes 6.8, 6.9 & 6.12 Example 1a) The pint P(4, 7) is n the terminal arm f an angle θ in standard psitin. a) Determine the distance r frm the rigin t P. Outcme 6.8 b) Determine the primary trignmetric ratis f θ. Outcme 6.9 r P(x, y) c) Determine the measure f θ t the nearest degree. Outcme

6 Outcmes 6.8, 6.9 & 6.12 Example The angle θ is in standard psitin in Quadrant I. ; what are sin θ and tan θ? r P(x, y) 6

7 Outcmes 6.8, 6.9 & 6.12 Try this ne... The pint P(3, 4) is n the terminal arm f an angle in standard psitin. a) Determine the distance r frm the rigin t P. P(x, y) b) Determine the primary trignmetric ratis f θ. r c) Determine the measure f θ t the nearest degree. 7

8 Outcmes 6.10, 6.12 & 6.15 Directinal Ntatin: Bearings & Headings Trignmetry is essential t navigatin. The scale n a directinal cmpass shw rute bearings r headings. As a result, directins can be described and written 3 ways using bearings & headings! Bearings are written using ONLY an angle between Bearings start at due Nrth always, cntinue clckwise & are measured in degrees written with three digits! Example: A bearing f 250 describes a directin in quadrant 3 n the grid. 8

9 Outcmes 6.10, 6.12 & 6.15 Example 9

10 Outcmes 6.10, 6.12 & 6.15 Headings are written using the fur cmpass pints f nrth, suth, east and west. Each f these directins describe tw quadrants n a grid. As a result, they are always angles between 0 90 and must include the tw directins f the quadrant. The first letter is the directin where yu start frm, and the secnd letter is the directin yu are headed twards. Hw far yu travel (the angle) is written between the letters. Example: N50 E This means yu start at due nrth n the grid and travel 50 twards east. This directin can als be written the ppsite way reversing the directins = E40 N and subtracting the angle frm 90 t find the new angle: 10

11 Outcmes 6.10, 6.12 &

12 Outcmes 6.10, 6.12 & 6.15 Yur turn, try writing these directins... Hint: It is always best t draw the directins n a grid t get an accurate view f the situatin! 1. Yu travel 68 suth f west. 2. Yu travel 72 nrth f east. 3. Yu travel 25 west f nrth. 4. Yu travel 13 east f suth. 5. Yu travel 64 nrth f west. 6. Yu travel 39 suth f east. 12

13 Outcmes 6.10, 6.12, 6.15 & 6.16 Example An aircraft made an emergency landing 200 km frm an airprt. Its heading frm the airprt was E50 N. The land-based rescue team has t travel east then nrth t get t the aircraft. T the nearest kilmeter, hw far shuld the team travel in each directin? 13

14 Outcmes 6.10, 6.12, 6.15 & 6.16 Example A frest ranger sees smke rising frm a pint that lies in a directin 40 N f E. She estimates that the distance frm the ranger statin is abut 30 km. The firefighters at the ranger statin have t travel east then nrth t get t the fire. T the nearest kilmeter, hw far shuld the firefighters travel in each directin? 14