Math 0993
Learning Centre
Advanced Trigonometry
SINE AND COSINE IDENTITIES
SPECIAL TRIANGLES
sin θ = sin (θ + 2kπ), k is an integer
sin (−θ) = −sin θ *
sin (θ ± π) = − sin θ *
sin (π − θ) = sin θ *
2
1
45°
sin (α ± β) = sin α cos β ± cos α sin β
cos (α ± β) = cos α cos β ∓ sin α sin β
tan α ± tan β
tan (α ± β) =
1 ∓ tan α tan β
RECIPROCAL AND QUOTIENT IDENTITIES
sec θ = 1⁄cos θ, csc θ = 1⁄sin θ , cot θ = 1⁄tan θ
tan θ = sin θ⁄cos θ, cot θ = cos θ⁄sin θ
DOUBLE AND HALF ANGLE IDENTITIES
sin 2θ = 2 sin θ cos θ
cos 2θ = cos² θ − sin² θ
= 2 cos² θ − 1
= 1 − 2 sin² θ
2 tan θ
tan 2θ =
1 − tan² θ
PYTHAGOREAN IDENTITIES
sin² θ + cos² θ = 1
1 + cot² θ = csc² θ
1 + tan² θ = sec² θ
COFUNCTION IDENTITIES
sin (θ ± π2 ) = ± cos θ
cos (θ ± π2 ) =  sin θ
Any trig function of a
positive acute angle is
equal to the cofunction of
the complementary angle.
e.g. sin θ = cos ( π2 − θ)
or sin θ = cos (90° − θ)
PRINCIPAL VALUES OF INVERSE FNS
arcsin
−π⁄2 ≤ y ≤ π⁄2
arccos
0≤y≤π
+
−
− +
arccsc
−π⁄2 ≤ y < 0,
0 < y ≤ π⁄2
arcsec
0 ≤ y < π⁄2,
π⁄2 < y ≤ π
+
−
− +
arctan
−π⁄2 < y < π⁄2
The choice of + or − in
these three identities
depends on which
quadrant θ⁄2 lies in
1 − cos 2θ
2
sin θ⁄2 = ± 1 − cos θ
2
sin² θ =
1 + cos θ
2
cos² θ =
1 + cos 2θ
2
tan θ⁄2 = ± 1 − cos θ
1 + cos θ
tan² θ =
1 − cos 2θ
1 + cos 2θ
cos θ⁄2 = ±
RECIPROCALS AND COFUNCTIONS
cofunctions
1
SUM & DIFFERENCE IDENTITIES
** These are the same for other even functions.
sin θ
cos θ
tan θ
cot θ
sec θ
csc θ
60°
1
cos θ = cos (θ + 2kπ), k is an integer
cos (−θ) = cos θ **
cos (θ ± π) = −cos θ **
cos (π − θ) = −cos θ **
2
3
* These are the same for other odd functions.
reciprocals
30°
45°
=
sin θ
1 + cos θ
=
1 − cos θ
sin θ
IDENTITY FOR c sinx kθ + b cos kθ
c sin kθ + b cos kθ = a sin (kθ + β)
where:
+
−
arccot − +
0<y<π
© 2016 Vancouver Community College Learning Centre.
Student review only. May not be reproduced for classes.
b2 + c 2
b
β = tan−1
c
a=
a
β
b
c
Authoredby
byEmily
Gordon
Wong
Simpson
ANGULAR MOTION
s
s = arc length
same units
r = radius
θ = central angle (rad)
ω = angular speed
v = linear speed
(at a point on rim)
t = time
s
r
θ
ω=
t
v = rω
θ=
ANGLE SPECIFICATION TECHNIQUES
Directed Angles
y
• used for trig class
• +ve angles: start at the
θ
positive x-axis and go CCW
• −ve angles: start at the
positive x-axis and go CW
• angles higher than 360°
or 2π rad are possible
θ r
DEGREES, MINUTES AND SECONDS
1° = 60′
1′ = 60″
Heading, Bearing or Course
• used by airplanes and
N
0°
boats
• start at North, go CW
270°
• angle of arrival (coming W
into a destination) is 180°
away from the angle
200° 180°
leaving a location (e.g., if
S
angle of departure is 30°,
angle of arrival will be 210°)
POLAR COORDINATES
Polar  Rectangular (r, θ)  (x, y)
x = r cos θ
y = r sin θ
Rectangular  Polar (x, y)  (r, θ)
r = x2 + y2
θ = cos−1 2 x
x + y2
= sin−1
x
y
x2 + y2
VECTORS

if: u = (a, b)

v = (c, d)
= xr
= yr
Compass Directions
N N 50° E
• two forms:
(1) “N 50° E” starts
at North or South and
W
E
moves CW or CCW
65° south
eastward or westward
25° west
of west
of south
(2) “65° south of
S
west” starts at second
compass point and moves CW or CCW toward
the first compass point
• in form (2), every direction can be
expressed two different ways
 
then: u + v = (a + c, b + d)

u = a2 + b2
ANGLES OF ELEVATION AND DEPRESSION
angle of
elevation
angle of depression
SINE AND COSINE LAWS
Cosine Law (use with SAS or SSS)
a² = b² + c² − 2bc cos A
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C
A
b
c
Sine Law (use with AAS, SSA)
C
B
90°E
a
sin A sin B sin C
=
=
a
b
c
[Note: the SSA case may yield 0, 1 or 2 solutions.]
© 2016 Vancouver Community College Learning Centre.
Student review only. May not be reproduced for classes.
Wind Direction
N wind blows
• used in flying, weather
this way
reports
• wind direction is the
W
E
direction the wind is
S15°W
measured
blowing from, not to,
here
e.g., a west wind blows
S
from west to east
• find wind direction using above techniques;
draw the vector pointing 180° away from the
indicated angle
2