1.1 Trigonometry

1.1 Trigonometry

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.

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.

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

Initial side

Initi

al si

de

Vertex

Vertex

Term

inal

sid

e

Terminal side

This represents a positive angle

This represents a negative angle

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

There are two ways to measure angles…

Degrees

Radians

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.

arc

radius

1 Radian

Length of the arc is equal to the length of the radius.

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.

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.

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.

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


= 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)


= -90

positive coterminal angle: -90 + 360 = 270 negative coterminal angle: - 90 – 360 = - 470

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.


= /7

positive coterminal angle: /7 + 2 = /7 + 14/7 = 15/7 rad negative coterminal angle: /7 - 2 = /7 - 14/7 = -13/7 rad


= -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

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).

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

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).

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/

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.

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

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

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



9/2 radians

810

9 radians 180 degrees = 2 radians 1620 degrees =

2



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.)

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!

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

Examples

r= 14 inches

3 radians

s = ?

s = r s = (14)(3)s = 42 inches

Picture not drawn to scale.

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.

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1.1 Trigonometry