1.1 trigonometry
DESCRIPTION
1.1 Trigonometry. Vocabulary:. Angle – created by rotating a ray about its endpoint. Initial Side – the starting position of the ray. Terminal Side – the position of the ray after rotation. Vertex – the endpoint of the ray. - PowerPoint PPT PresentationTRANSCRIPT
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1.1 Trigonometry
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Vertex – the endpoint of the ray.
Vocabulary:Angle – created by rotating a ray about its endpoint.
Initial Side – the starting position of the ray.
Terminal Side – the position of the ray after rotation.
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Initial side
Initi
al si
de
Vertex
Vertex
Term
inal
sid
e
Terminal side
This arrow means that the rotation was in a counterclockwise direction.
This arrow means that the rotation was in a clockwise direction.
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Positive Angles – angles generated by a counterclockwise rotation. Negative Angles – angles generated by a clockwise rotation. We label angles in trigonometry by using the Greek alphabet. - Greek letter alpha - Greek letter beta - Greek letter phi - Greek letter theta
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Initial side
Initi
al si
de
Vertex
Vertex
Term
inal
sid
e
Terminal side
This represents a positive angle
This represents a negative angle
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Standard Position – an angle is in standard position when its initial side rests on the positive half of the x-axis.
Positive angle in standard position
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There are two ways to measure angles…
Degrees
Radians
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Degrees:• There are 360 in a complete circle.• 1 is 1/360th of a rotation.
Radians:• There are 2 radians in a complete circle.• 1 radian is the size of the central angle when the radius of the circle is the same size as the arc of the central angle.
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arc
radius
1 Radian
Length of the arc is equal to the length of the radius.
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Coterminal angles – two angles that share a common vertex, a common initial side and a common terminal side.
Examples of Coterminal Angles
and are coterminal angles because they share the same initial side and same terminal side.
Coterminal angles could go in opposite directions.
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Examples of Coterminal Angles
and are coterminal angles because they share the same initial side and same terminal side.
Coterminal angles could go in the same direction with multiple rotations.
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Finding coterminal angles of angles measured in degrees:
Since a complete circle has a total of 360, you can find coterminal angles by adding or subtracting 360 from the angle that is provided.
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Example:Find two coterminal angles (one positive and one negative) for the following angles.
= 25
positive coterminal angle: 25 + 360 = 385 negative coterminal angle: 25 – 360 = - 335
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Example:Find two coterminal angles (one positive and one negative) for the following angles.
= 725
positive coterminal angle: 725 + 360 = 1085 (add a rotation) or 725 – 360 = 365 (subtract a rotation) or
725 – 360 – 360 = 5 (subtract 2 rotations)negative coterminal angle: 725 – 360 – 360 – 360 = - 355 (must subtract 3 rotations)
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Example:Find two coterminal angles (one positive and one negative) for the following angles.
= -90
positive coterminal angle: -90 + 360 = 270 negative coterminal angle: - 90 – 360 = - 470
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Finding coterminal angles of angles measured in radians:
Since a complete circle has a total of 2 radians you can find coterminal angles by adding or subtracting 2 from the angle that is provided.
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Example:Find two coterminal angles (one positive and one negative) for the following angles.
= /7
positive coterminal angle: /7 + 2 = /7 + 14/7 = 15/7 rad negative coterminal angle: /7 - 2 = /7 - 14/7 = -13/7 rad
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Example:Find two coterminal angles (one positive and one negative) for the following angles.
= -4/9
positive coterminal angle:-4/9 +2 = -4/9 + 18/9 =14/9 rad negative coterminal angle:-4/9 -2 =-4/9 - 18/9 =-22/9 rad
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Complementary angles – two positive angles whose sum is 90 or two positive angles whose sum is /2.
To find the complement of a given angle you subtract the given angle from 90 (if the angle provided is in degrees) or from /2 (if the angle provided is in radians).
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Example:Find the complement of the following angles if one exists. = 29
complement = 90 – 29 = 61
= 107
.
complement = 90 – 107 = none(No complement because it is negative)
= /5
complement = /2 - /5 = 5/10 - 2/10 = 3/10
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Supplementary angles – two positive angles whose sum is 180 or two positive angles whose sum is .
To find the supplement of a given angle you subtract the given angle from 180 (if the angle provided is in degrees) or from (if the angle provided is in radians).
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Example:Find the supplement of the following angles if one exists. = 29
supplement = 180 – 29 = 151
= 107supplement = 180 – 107 = 73
= /5
supplement = - /5 = 5/5 - /5 = 4/
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We have to become comfortable working with both forms of measuring angles.
Therefore, MEMORIZE the following:
Degrees Radians Degrees Radians0 0 radians 90 /2 radians
30 /6 radians 180 radians45 /4 radians 270 3/2 radians60 /3 radians 360 2 radians
We will memorize more, very, very soon.
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Manually Converting from Degrees to Radians:
Multiply the given degrees by radians/180
Example:Convert the following degrees to radians
135
3 radians 4
135 degrees radians = 1 180 degrees 135 radians =
180
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Multiply the given degrees by radians/180
Example:Convert the following degrees to radians
540
3 radians 1
540 degrees radians = 1 180 degrees 540 radians =
180
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Manually Converting from Radians to Degrees:
Multiply the given radians by 180/ radians
Example:Convert the following radians to degrees.
-/3 radians
-60
- radians 180 degrees = 3 radians -180 degrees =
3
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Multiply the given radians by 180/ radians
Example:Convert the following radians to degrees.
9/2 radians
810
9 radians 180 degrees = 2 radians 1620 degrees =
2
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Multiply the given radians by 180/ radians
Example:Convert the following radians to degrees.
2
114.59
2 radians 180 degrees = 1 radians 360 degrees =
2(if you don’t see the degree symbol, then the angle measure is automatically believed to be a radian.)
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Tomorrow, we will look at your individual calculators and show you how to do these conversions via those calculators.
BRING YOUR OWN SCIENTIFIC
CALCULATOR TOMORROW!
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Finding Arc Length:
•The following formula is used to determine arc length: s = r
arc length radiusMeasure of the central angle in radians.
must have the same units of measure
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Examples
r= 14 inches
3 radians
s = ?
s = r s = (14)(3)s = 42 inches
Picture not drawn to scale.
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Examples
r= ?
30
s =9 cm
s = r 9 = (r)(/6)r = 54/ cm 17.19 cm
Picture not drawn to scale.
You must convert 30 to radians.