math sl1 - santowski1 t3.1 - radian measure ib math sl1 - santowski
DESCRIPTION
Math SL1 - Santowski3 (A) Angles in Standard Position 3 Angles in standard position are defined as angles drawn in the Cartesian plane where the initial arm of the angle is on the x axis, the vertex is on the origin and the terminal arm is somewhere in one of the four quadrants on the Cartesian plane To form angles of various measure, the terminal arm is simply rotated through a given angleTRANSCRIPT
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Math SL1 - Santowski 1
T3.1 - Radian Measure
IB Math SL1 - Santowski
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Fast Five
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Evaluate sin(50°) illustrate with a diagram Evaluate sin(130°) illustrate with a diagram
Evaluate sin(230°) illustrate with a diagram?? Evaluate sin(320°) illustrate with a diagram?? Evaluate sin(770°0 illustrate with a diagram??
Evaluate sin(-50°) illustrate with a diagram??
Use your calculator and graph the function f(x) = sin(x) on the domain -720° < x < 720°
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(A) Angles in Standard Position
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Angles in standard position are defined as angles drawn in the Cartesian plane where the initial arm of the angle is on the x axis, the vertex is on the origin and the terminal arm is somewhere in one of the four quadrants on the Cartesian plane
To form angles of various measure, the terminal arm is simply rotated through a given angle
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(A) Angles in Standard Position – Interactive Applet
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Go to the link below and work through the ideas presented so far with respect to angles in standard position
Angles In Trigonometry from AnalyzeMath
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(B) Radians We can measure angles in several ways - one
of which is degrees Another way to measure an angle is by means
of radians One definition to start with an arc is a
distance along the curve of the circle that is, part of the circumference
One radian is defined as the measure of the angle subtended at the center of a circle by an arc equal in length to the radius of the circle
Now, what does this mean?
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(B) Radians
If we rotate a terminal arm (BC) around a given angle, then the end of the arm (at point B) moves along the circumference from A to B
If the distance point B moves is equal in measure to the radius, then the angle that the terminal arm has rotated is defined as one radian
If B moves along the circumference a distance twice that of the radius, then the angle subtended by the arc is 2 radians
So we come up with a formula of = arc length/radius = s/r
Radius
arc
AC
B
angle
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(C) Converting between Degrees and RadiansIf point B moves around the entire circle, it has revolved or rotated 360°
Likewise, how far has the tip of the terminal arm traveled? One circumference or 2r units.
So in terms of radians, the formula is = arc length/radius = s/r = 2 r/r = 2 radians
So then an angle of 360° = 2 radians or more easily, an angle of 180° = radians
Radius
arc
AC
B
angle
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(C) Converting from Degrees to Radians Our standard set of first quadrant angles include 0°,
30°, 45°, 60°, 90° and we now convert them to radians:
We can set up equivalent ratios as:
30° =
45° = 60° = 90° =
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(C) Converting from Degrees to Radians Our standard set of first quadrant angles include 0°,
30°, 45°, 60°, 90° and we now convert them to radians:
We can set up equivalent ratios as:
30°/ x radians = 180°/ radians Then x = /6 radians
45°/x = 180°/ x = /4 radians 60°/x = 180°/ x = /3 radians 90°/x = 180°/ x = /2 radians
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(D) Converting from Radians to Degrees Let’s work with our second quadrant
angles with our equivalent ratios:
2/3 radians 3/4 radians 5/6 radians
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(D) Converting from Radians to Degrees Let’s work with our second quadrant
angles with our equivalent ratios: 180°/ = x / (2/3)
x = (2/3)(180/) = 120°
180°/ = x / (3/4) x = (3/4)(180/) = 135°
180°/ = x / (5/6) x = (5/6)(180/) = 150°
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(E) Table of Equivalent Angles We can compare the measures of important angles in
both units on the following table:
0 ° 90° 180° 270° 360°
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(E) Table of Equivalent Angles We can compare the measures of important angles in
both units on the following table:
0 ° 90° 180° 270° 360°
0 rad /2 rad rad 3/2 rad 2 rad
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(E) Table of Equivalent Angles
We can compare the measures of important angles in both units on the following table:
30 45 60 120 135 150 210 225 240 300 315 330
/6 /4 /3 2 /3 3/4 5/6 7/6 5/4 4/3 5/3 7/4 11/6
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(F) Area of a Sector Recall the area of a circle formula: A = r2
So how would you find the area of: (i) half a circle (ii) a quarter of a circle (iii) one tenth of a circle (iv) the area swept out by the terminal arm as
it rotates 36 degrees? (v) the area swept out by the terminal arm as
it rotates 90 degrees? (vi) the area swept out by the terminal arm as
it rotates /2 radians?
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(F) Area of a Sector CONCLUSION from previous slide You can
work out the Area of a Sector by comparing its angle to the angle of a full circle.
Note: I am using radians for the angles.
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(G) Arc Length Recall the circumference of a circle formula: A
= 2r So how would you find the arc length of: (i) half a circle (ii) a quarter of a circle (iii) one tenth of a circle (iv) the arc length traveled by the terminal arm
as it rotates 36 degrees? (v) the arc length traveled by the terminal arm
as it rotates 90 degrees? (vi) the arc length traveled by the terminal arm
as it rotates /2 radians?
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(G) Arc Length Arc Length of Sector or Segment By the same reasoning, the arc length
(of a Sector or Segment) is Arc Length "L" = θ × r
= (θ × π/180) × r (if θ is in degrees)
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(H) Internet Links Topics in trigonometry: Radian me
asure from The Math Page Measurement of angles from David
Joyce, Clark University Radians and Degrees - on-line Mat
h Lesson from TV
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(I) Homework
HW - Ex 13B.2 (all), IB Packet #1 Nelson text, p442, Q2,3,5,6