Unit 5, Lesson 1

Angles and their measures are used in many real-world applications, such as

navigation & surveying

vertex

a fixed endpoint where the sides of an angle meet

standard position

an angle that has a vertex located at the origin of an x-y coordinate plane & one ray is on the positive x-axis

initial side

the ray of an angle in standard position that's on the positive x-axis

terminal side

the other ray of an angle in standard position

If the terminal side is rotated counterclockwise,

the angle is positive

If the terminal side is rotated clockwise,

the angle is negative

What is used to indicate the direction of rotation?

arrows

One full rotation in either direction is the equivalent of

360 degrees

Ancient Babylonians

Generally credited with dividing a circle, or one rotation, into 360 degrees using their number system, which is based on units of 60

There are very few other numbers as small as 360 that are

divisible by so many factors; therefore, 360 is a convenient number because it can be used to divide a rotation into an equal number of parts

One-sixtieth of a rotation is equal to

one unit or one degree

One-half of a rotation is equal to

180 degrees

One-sixth of a rotation is equal to

60 degrees

One-twelfth of a rotation is equal to

30 degrees

The unit measure, degree, can be subdivided into

60 equal parts known as minutes, each part being one-sixtieth of a degree

minutes

make up a degree; each one is one-sixtieth of a degree; 1 degree = 60'; 1' = 60''; each minute is split up into 60 parts, each part being one-sixtieth of a minute

seconds

make up a minute; each one is one-sixtieth of a minute; 1' = 60''

The words minute and second used in this context have no immediate connection to

how they're usually used as amounts of time; instead they measure the size of an angle

How can you change the degree form into decimal form?

multiply the given minutes by one-sixtieth and the given seconds by one-sixtieth twice; simplify and round to the nearest ten-thousandths place

How to Convert Degrees from Decimal Form into Minutes and Seconds

1. Determine how many full degrees there are2, Set up a ratio to solve for the remaining degrees; for minutes, use ratio x degrees times (60 minutes/1 degree)3. Determine how many full minutes there are4. Set up a ratio to solve for the remaining minutes; for seconds, use ratio x minutes times (60 seconds/1 minute)5. Simplify and round as necessary; put final answer in degree-minute-second form

quadrantal angles

angles in standard position whose terminal side coincides with one of the coordinate axes; measure 90 degrees, 180 degrees, 270 degrees, and 360 degrees

The measure of a quadrantal angle can be determined by

multiplying the number of rotations by 360 degrees

If the rotation is clockwise, multiply by

-360 degrees

If the rotation is counterclockwise, multiply by

360 degrees

coterminal angles

two angles in standard position that have the same terminal side

Coterminal angles can be found by

adding or subtracting 360 degrees from the original angle due to simple rotation of the circle

reference angle

the positive acute angle formed between the terminal side of an angle and the x-axis

The use of reference angles is

a way to simplify the calculation of the values of trigonometric functions at various angles

How to Determine a Reference Angle

1. Find the coterminal angle of the given angle2. Determine what quadrant it is in3. Use the formula for the quadrant to find the reference angle

Reference Angle Rule

For any angle a, 0 degrees < a < 360 degrees, its reference angle a' is defined by:a. a, when the terminal side is in Quadrant I,b. 180 degrees - a, when the terminal side is in Quadrant II,c. a - 180 degrees, when the terminal side is in Quadrant III, andd. 360 degrees - a, when the terminal side is in Quadrant IVIf the measure of a is greater than 360 degrees or less than 0 degrees, it can be associated with a coterminal angle of positive measure between 0 degrees and 360 degrees