Trigonometry Vocab
For the negative angle -120°, add 360° to obtain a coterminal angle
-120° + 360° = 240°
CALCULATOR!!!!!!!!!!!!!! Change degrees to degrees, minutes, and seconds.
1. Type in the degrees 2. Press 2nd, Angle, DMS, Enter 3. Round of the seconds because there are NO. DECIMALS!
supplement of an angle
in degrees: 180° - angle in radians: π - angle
complement of an angle
in degrees: 90° - angle in radians: π/2 - angle
s
is the arc length
θ
is the central angle in radians
r
is the radius of the circle
a sector of a circle
is the region bounded by two radii of the cricle and their intercepted arc.
the ratio s/r has no units
it is simply a real number
one degree (1°)
a rotation of 1/360 of a complete revolution about the vertex
radians
a type of measure especially useful in calculus
To find an angle that is coterminal to a given angle θ
add or subtract 360° (one revolution)
coterminal angles
angle + (n)360° where n is an integer
coterminal angles
angles that have the same initial and terminal sides angles that start and stop in the same place but have a different measure
one full revolution
arc length of s = 2πr 2π is about 6.28 just over six radian lengths
Angles are labeled with Greek letters α (alpha), β (beta), and θ (theta)
as well as uppercase A, B, and C
acute angles
between 0° and 90°
obtuse angles
between 90° and 180°
negative angles
clockwise rotation
Two positive angles α and β are
complementary (complements of each other) if their sum is 90°.
positive angles
counterclockwise rotation
angle
determined by rotating a ray (half-line) about its endpoint
Quadrant I
Positive Degrees: 0° < θ < 90° Negative Degrees: -270° < θ < -360° Positive Radians: 0 < radian < π/2 Negative Radians: -3π/2 < radian < 2π
Quadrant III
Positive Degrees: 180° < θ < 270° Negative Degrees: -90° < θ < -180° Positive Radians: π < radian < 3π/2 Negative Radians: -π/2 < radian < -π
Quadrant IV
Positive Degrees: 270° < θ < 360° Negative Degrees: 0° < θ < -90° Positive Radians: 3π/2 < radian < 2π Negative Radians: 0 < radian < -π/2
Quadrant II
Positive Degrees: 90° < θ < 180° Negative Degrees: -180° < θ < -270° Positive Radians: π/2 < radian < π Negative Radians: -π < radian < -3π/2
To convert radians to degrees
Radians x 180°/π
CALCULATOR!!!!!!!!!!!!!! Change degrees, minutes, and seconds to degrees.
1. Type the degree 2. Press 2nd, Angle, and the ° symbol, which is 1 3. Type the minute 4. Press 2nd, Angle, and the ' symbol which is 2 5. Type the seconds 6. Press Alpha, the + button, and then Enter
Solving Right Triangles
1. given on side of a right triangle and one of the acute triangles, find one of the other sides OR 2. given two sides; find one of the acute angles OR 3. find all of the remaining parts
For the positive angle 135°, subtract 360° to obtain a coterminal angle.
135° - 360° = -225°
half revolution (counterclockwise)
180°
1 radian
180°/π
circumference
2πr units
1 revolution
360°
2πr
360°
2πradians
360°
circumference
360°
full revolution (counterclockwise)
360°
For the positive angle 360° subtract 360° to subtract 360° to obtain a coterminal angle.
390° - 360° = 30°
quarter revolution (counterclockwise)
90°
area of a sector of a circle
A = 1/2r^2θ r is the radius of the circle. θ is the central angle IN RADIANS
To convert degrees to radians
Degrees x π/180°
A given angle θ has infinitely many coterminal angles
Example: θ = 30° is coterminal with 30° + n(360°) where n is an integer
trigonometry
Greek for "measurement of triangles"
Angles between 0° and 360°
lie in each of the quadrants
definition of radian
one radian is the measure of a central angle θ that intercepts an arc s equal in length of the radius r of the circle
terminal side
position after rotation
arc length
s = rθ
To evaluate function with a calculator
set the calculator to the desired mode of measurement (degree or radian)
RECIPROCALS
sine (sin) <--> cosecant (csc) cosine (cos) <--> secant (sec) tangent (tan) <--> cotangent (cot)
COFUNCTIONS
sine <--> cosine tangent <--> cotangent secant <--> cosecant
Two positive angles α and β are
supplementary (supplements of each other) if their sum is 180°.
measure of an angle
the amount of rotation from the initial side to the terminal side
standard position of an angle
the angle fits in the coordinate system the initial side coincides with the positive x-axis the vertex is at origin
angle of depression
the angle from the horizontal downward to an object
angle of elevation
the angle from the horizontal upward to an object
vertex
the endpoint of the ray
degrees
the most common unit of angle measure (denoted by the symbol, °)
initial side
the starting position of the ray
If 0° < θ < 90° (θ lies in the first quadrant)
then the value of each trigonometric function is POSITIVE
The reciprocal identities for sine, cosine, and tangent can be used
to evaluate the cosecant, secant, and cotangent functions with a calculator
To measure angles, it is convenient
to mark degrees on the circumference of a circle
To evaluate csc(π/8)
use the reciprocal function: csc(π/8) = 1/sin(π/8)
To evaluate cotangent, secan, and co
use the x^-1 wither their respective reciprocal functions sine, cosine, and tangent.
central angle
vertex is the center of the circle
arc length = radius
when θ = 1 radian
Algebraically, θ - s/r
where θ is measured in radians. (Note: θ = 1 when s = r)
1°
π/180°