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