Math 150: 4.1 Radian and Degree Measure and 4.3 Right Triangle Trigonometry
1. Angle Measure
a) arbitrarily defined angles (fractions of a circle): degrees (1/360), gradians (1/400)
b) radians: an angle formed by a sector of a circle whose arclength equals its radius
2 rad
3 rad
1 rad
0.28 rad
4 rad
5 rad
6 rad
c) DMS: degrees (like hours), minutes (1/60 of a degree), seconds (1/60 of a minute)
2. Circle Formulas s is arclength in m, etc. and θ is angle in rad; v is linear speed in m/s and ω is angular speed in rad/s
a) Arclength: s = θr and Circumference: C = 2πr (a special case where θ = 2π )
b) Area: Asector =
1
2
θr 2 (of a sector) and A = πr 2 (of a circle, where θ = 2π )
(
c) Constant speed (distance/time): Linear v =
d) Linear and Angular Speed:
© Raelene Dufresne 2013
s = θr
t
⇒
s
t
=
θ
t
s
t
) and Angular (ω = )
θ
3 rad
π rad
2π rad
6 rad
t
⋅ r ⇒ v = ωr
1 of 5
Math 150: 4.1 Radian and Degree Measure and 4.3 Right Triangle Trigonometry
3. Terms for Angles
a) Initial arm and terminal arm:
b) Acute ( 0o < θ < 90o ), right ( θ = 90o ), obtuse ( 90o < θ < 180o ), straight ( θ = 180o ) and reflex ( 180o < θ < 360o ) angles
c) Complementary Angles (two angles that sum to 90o ) and Supplementary Angles (two angles that sum to 180o )
d) Positive angle
(ccw: counter-clockwise)
Negative angle
(cw: clockwise)
e) Quadrantal angles (lie ON an axis: 0o , 90o , 180o , 270o , 360o and Nonquadrantal angles (lie IN a quadrant)
f) Standard Position (initial arm at 0o and measuring ccw) vs. Compass Position (N, NE, NW, 30o west of south, etc.)
© Raelene Dufresne 2013
2 of 5
Math 150: 4.1 Radian and Degree Measure and 4.3 Right Triangle Trigonometry
g) Principal and Coterminal Angles: Two or more angles that are coterminal share the same initial and terminal arms.
The angle that lies in the first rotation of the terminal arm (between 0o and 360o ) is the principal angle.
(
)
(
)
θ1 = 120o
θ2 = 120o + 360o = 480o
θ3 = 120o + 2 360o = 840o
θ 4 = 120o − 1 360o = −240o
Principal Angle
Positive Coterminal Angle
Positive Coterminal Angle
Negative Coterminal Angle
h) Reference (or Related Acute) Angles: The reference or related acute angle is an acute angle that is related to a
given quadrant (I, II, III or IV) angle. To get the reference angle, find the angle between the given angle and the
nearest x – axis angle ( 0o , 180o or 360o ). For example, the angles 30o , 150o , 210o and 330o all lie in different
quadrants but have the same reference angle of 30o ; note that the reference angle is acute (between 0o and 90o ).
θI = 30o
θII = 150o
θIII = 210o
θIV = 330o
These four principal angles all have a reference angle of 30 degrees: θref = 30o .
© Raelene Dufresne 2013
3 of 5
Math 150: 4.1 Radian and Degree Measure and 4.3 Right Triangle Trigonometry
Note: Only nonquadrantal angles have reference angles. The quadrantal angles ( 0o , 90o , 180o , 270o , 360o , etc.) do NOT
have a reference angle, since they are either 0o or 90o from the nearest x – axis; reference angles are ACUTE
( 0o < θ ref < 90o ).
4. Anatomy of a Right Triangle and the 6 Trigonometric Functions:
THREE SIDES è SIX RATIOS:
opposite
opposite to θ
hypotenuse
hypotenuse
θ
hypotenuse
adjacent to θ
opposite
,
,
adjacent
hypotenuse
hypotenuse
adjacent
opposite
,
adjacent
adjacent
,
opposite
Therefore there are 6 trigonometric functions (or ratios), defined as follows:
Primary Trig Ratio
sin θ =
cos θ =
opposite
hypotenuse
adjacent
hypotenuse
tan θ =
opposite
adjacent
© Raelene Dufresne 2013
Reciprocal Trig Ratio
csc θ =
secθ =
hypotenuse
opposite
hypotenuse
cot θ =
adjacent
adjacent
opposite
Memory aid for remembering the three main trig ratios:
SOH – CAH - TOA
Full names of trig functions:
sin à sine
cos à cosine
tan à tangent
csc à cosecant
sec à secant
cot à cotangent
NOTE: { sin−1 θ , the inverse of sin θ } ≠ {
1
sin θ
, the reciprocal of sin θ }
4 of 5
Math 150: 4.1 Radian and Degree Measure and 4.3 Right Triangle Trigonometry
5. Fundamental Trig Identities (an equation that is true
∀ values of the variable in the domain of the functions):
6. Two Special Triangles
(to evaluate trig ratios of 30o , 45o or 60o )
Quotient Identities
a) tan θ =
sin θ
b) cot θ =
cos θ
PROOF:
sinθ
RS =
cosθ
=
=
=
cos θ
PROOF:
sinθ
opp
hyp
adj
hyp
opp hyp
⋅
hyp adj
opp
adj
= tan θ
= LS
∴ LS = RS QED
7. Pythagorean Triples:
Any Proportion
or
3k , 4k ,5k
a) 3, 4, 5:
32 + 42 = 52
b) 5, 12, 13:
5 + 12 = 13
or
5k ,12k ,13k
c) 7, 24, 25:
72 + 242 = 252
or
d) 8, 15, 17:
82 + 152 = 172
or
i.e.,
6, 8, 10
because 62 + 82 = 102
i.e.,
1, 12/5, 13/5
⎛ 12 ⎞
⎛ 13 ⎞
because 1 + ⎜ ⎟ = ⎜ ⎟
⎝ 5⎠
⎝ 5⎠
7k ,24k ,25k
i.e.,
21, 72, 75
because 212 + 722 = 752
8k ,15k ,17k
i.e.,
40, 75, 85
because 402 + 752 = 852
2
2
© Raelene Dufresne 2013
2
2
2
2
5 of 5