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6.1 Angles in Standard Position In Quadrant 1.notebook 1 May 24, 2015 Unit 6 Trigonometry 6.1 Angles in Standard Position in Quadrant 1
Transcript of 6.1 Angles in Standard Position In Quadrant 1.notebookhjcmath.pbworks.com/w/file/fetch/96684969/6.1...
6.1 Angles in Standard Position In Quadrant 1.notebook
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Unit 6 Trigonometry6.1 Angles in Standard Position
in Quadrant 1
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For a point on the coordinate plane, when an angle θ isbetween 0o and 360o, the angle is measured counterclockwise from the positive x-axis, the initial arm. This angle is said to be in ________ _________.The line joining the point and the origin is called the _____________ ____________ . The point P is the _____________ ____________.
standard
terminal arm
position
terminal point
6.1 Trigonometry: Angles in Standard PositionOutcomes
6.1, 6.2 & 6.6 Determine & sketch the angle in standard position given a point P(x, y) on the terminal arm of the angle & determine the reference angle for its angle in standard position.
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Trigonometry Key Terms:
x
y
The Cartesian Plane:has four quadrants that run counter clockwise starting at the top right quadrant which is positive (x, y).
Initial Arm: the positive xaxis; the starting point of the rotation.
Terminal Arm: the stopping point of the rotation.
Coterminal Angle: Two or more angles in standard position that have the same terminal arm.
Reference Angle: The acute angle between the terminal arm and the xaxis.
General Solution: All possible solutions as a function of “n”.
II
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- r in terms of x and y
- the value of θ in terms of x and y
- the x-coordinate of P in terms of r and θ.
- the y-coordinate of P in terms of r and θ.
P(x, y)
r
Using the diagram below, write an expression for:
Outcome 6.1
Angles in Standard Position in Quadrant ISketch an angle in standard position, given the measure of the angle.
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ç üExample 1a) The point P(4, 7) is on the terminal arm of an angle θ in standard position.a) Determine the distance r from the origin to P.
b) Determine the primary trigonometric ratios of θ.
c) Determine the measure of θ to the nearest degree.
P(x, y)
r
Outcome 6.8
Outcome 6.9
Outcome 6.12
Outcomes 6.8, 6.9 & 6.12
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The angle θ is in standard position in Quadrant I.; what are sin θ and tan θ?
P(x, y)
r
Outcomes 6.8, 6.9 & 6.12 Example
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The point P(3, 4) is on the terminal arm of an angle in standard position.
a) Determine the distance r from the origin to P.
b) Determine the primary trigonometric ratios of θ.
c) Determine the measure of θ to the nearest degree.
ç ü
Try this one...
P(x, y)
r
Outcomes 6.8, 6.9 & 6.12
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Directional Notation: Bearings & Headings
Bearings are written using ONLY an angle between 0 360 . Bearings start at due North always, continue clockwise & are measured in degrees written with three digits!
Example: A bearing of 250 describes a direction in quadrant 3 on the grid.
o
o o
Trigonometry is essential to navigation. The scale on a directional compass show route bearings or headings. As a result, directions can be described and written 3 ways using bearings & headings!
Outcomes 6.10, 6.12 & 6.15
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Outcomes 6.10, 6.12 & 6.15 Example
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Headings are written using the four compass points of north, south, east and west. Each of these directions describe two quadrants on a grid. As a result, they are always angles between 0 90 and must include the two directions of the quadrant.
The first letter is the direction where you start from, and the second letter is the direction you are headed towards. How far you travel (the angle) is written between the letters.
Example: N50 E This means you start at due north on the grid and travel 50 towards east. This direction can also be written the opposite way reversing the directions = E40 Nand subtracting the angle from 90 to find the new angle:
o o
o
o
o
o
Outcomes 6.10, 6.12 & 6.15
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Outcomes 6.10, 6.12 & 6.15
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Your turn, try writing these directions...
1. You travel 68 south of west.o
2. You travel 72 north of east.
3. You travel 25 west of north.
4. You travel 13 east of south.
5. You travel 64 north of west.
6. You travel 39 south of east.
o
o
o
o
o
Hint: It is always best to draw the directions on a grid to get an accurate view of the situation!
Outcomes 6.10, 6.12 & 6.15
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An aircraft made an emergency landing 200 km from an airport. Its heading from the airport was E50oN. The land-based rescue team has to travel east then north to get to the aircraft. To the nearest kilometer, how far should the team travel in each direction?
ç ü
Outcomes 6.10, 6.12, 6.15 & 6.16 Example
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A forest ranger sees smoke rising from a point that lies in a direction 40o N of E. She estimates that the distance from the ranger station is about 30 km. The firefighters at the ranger station have to travel east then north to get to the fire. To the nearest kilometer, how far should the firefighters travel in each direction?
ç ü
Outcomes 6.10, 6.12, 6.15 & 6.16 Example
