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REFERENCE SHEET FOR TRIGONOMETRY
Chapters 7, 8, 9, 10
Trigonometric functions of acute angle :
opposite sin y


adjacent cos x
sin 
opposite
y

hypotenuse r
cos 
adjacent
x

hypotenuse r
csc 
hypotenuse
1
r


opposite
sin y
sec 
hypotenuse
1
r
adjacent
1
cos x


cot  



adjacent
cos x
opposite tan sin y
tan 
Cofunctions of complementary angles are equal. The two acute angles of a right triangle add to 90; thus
they are complementary. Given 2 complementary angles,  and , the cofunctions are:
sin  cos   cos  90   
tan  cot   cot  90   
sec  csc   csc  90   
cos  sin   sin  90   
cot   tan   tan  90    csc  sec   sec  90   
Identities: (1) sin2 + cos2 = 1; (2) sec2 - tan2 = 1; (3) csc2 - cot2 = 1
Quandrantal angles are angles whose terminal side lies on the x- or y-axis, such as 0=0, 90=/2, 180 = ,
270 = 3/2, and 360=2. Their trig function values will always be 0, 1, or undefined.
The trigonometric functions are periodic because, if we add an integral multiple of  or 2 to , the value of the
function is unchanged. All the functions have a period of 2 except the tangent and cotangent, which have a
period of . Thus, sin ( + 2) = sin , tan ( + ) = tan , etc.
DOMAIN AND RANGE
Function
Symbol
Domain
Range
Sine
[-1, 1]
f() = sin 
(-, )
Cosine
[-1, 1]
f() = cos 
(-, )

Tangent
f() = tan 
{xx  odd multiples of /2 (90)
(-, )
Cosecant
f() = csc 
{xx  integral multiples of  (180)
(-, -1]  [1, )

Secant
f() = sec 
{xx  odd multiples of /2 (90)
(-, -1]  [1, )
Cotangent
f() = cot 
{xx  integral multiples of  (180)
(-, )
The sine function, like all the trig functions except cosine and secant, are odd functions, thus symmetric to the
origin. It has x-intercepts at integral multiples of . The cosine function is even; thus it is symmetric to the yaxis. It has x-intercepts of multiples of /2.
The period of trig functions is affected by ω. The period of cos (ωx) = 2/ω. If ω > 1, the period will be
compressed horizontally by a factor of ω (omega). The amplitude (range) of A cos  is between -A and
A, inclusive. Sinusoidal Graphs: The graph of sin x = cos (x - /2). Because of this similarity, the graphs
of the two functions are called sinusoidal. Because the sine function is odd, sin(-ωx) = -sin(ωx). The cosine
function is even; therefore, cos(-ωx) = cos (ωx).
The tangent function has period , domain = {xx  odd multiples of /2}, range= (-, ). It has x-intercepts at
integral multiples of ; the y-intercept is 0. The cotangent function also has a period of  and its range = (-,
); but its domain = {xx  integral multiples of }. It has x-intercepts at odd multiples of /2 and no yintercept.
Given a function A sin (ωx – Φ) or A cos (ωx + Φ), the real number Φ/ω represents the phase shift; (ωx – Φ)
would be a sift of Φ/ω units to the right and (ωx + Φ) would be a sift of Φ/ω units to the left.
The inverse function can be used when we know the function value and want to find the angle . Since these
functions are not one-to-one, the domain must restricted in order to find an inverse function.
y = sin-1 x means x = sin y for –1 < x < 1 and -/2 < y < /2
y = cos-1 x means x = cos y for –1 < x < 1 and 0 < y < 
y = tan-1 x means x = tan y for – < x <  and -/2 < y < /2
Remember: [1] The domain of f-1(x) is the range of f(x) and vice versa.
[2] f-1(f(x) = x for every x in the domain of f and f(f-1(x) = x for every x in the domain of f-1.
y = sec-1 x means x = sec y for x > 1 and 0 < y < , y  /2
y = csc-1 x means x = csc y for x > 1 and -/2 < y < /2, y  0
y = cot-1 x means x = cot y for – < x <  and 0 < y < 
Basic Trigonometric Identities
sin
Quotient:
tan 
cos
cot  
sin2 + cos2 = 1;
sin(-) = -sin
csc(-) = -csc
Pythagorean:
Even-Odd:
cos
sin
Reciprocal:
csc 
tan2 + 1 = sec2;
cos(-) = cos
sec(-) = sec
1
sin
sec 
1
cos
cot 
1+ cot2 = csc2
tan(-) = -tan
cot(-) = -cot
1
tan
cos( + ) = cos cos - sin sin
[2] cos( - ) = cos cos + sin sin
[3] sin( + ) = sin cos + cos sin
[4]sin( - ) = sin cos - cos sin
tan  + tan 
tan  - tan 
[5]
[6]
tan( +  ) =
tan( -  ) =
1 - tan  tab 
1 + tan  tab 
Identities: [7]
[8]




cos  -   = sin 
sin  -   = cos
2

2

Double-Angle and Half-Angle Formulas
[1] sin (2) = 2 sin cos [2] cos(2) = cos2 - sin2 [3] cos(2) = 1 - 2sin2
[4] cos(2) = 2cos2 - 1
[5]
[6] 2 1 - cos  2 
[7]
[8]
1 + cos  2 
2tan 
2
1 - cos  2 
tan( 2 ) =
[9]
sin

2
=
sin  =
1 - tan 2
[10]
1 - cos
2

cos = 
2
cos  =
2
1 + cos
2
[11]
tan

2
=
tan 2 =
2
1 + cos  2 
1 - cos 1  cos
sin


1 + cos
sin
1  cos
PRODUCT-TO-SUM AND SUM-TO-PRODUCT FORMULAS
1
[1] sin  sin   cos      cos     
2
1
[3] sin  cos   sin      sin     
2
[5]
sin   sin   2sin

2
cos

[6]
2
cos   cos   2cos
1
[2] cos  cos   cos      cos     
2
[4] sin   sin   2sin

2
cos
 
2
[7]

2
cos

cos   cos   2sin
2

2
sin

2
Solving Oblique Triangles
Oblique triangles are triangles that do not contain a right angle; they have either 3 acute angles or 1 obtuse and
2 acute angles. Solving a triangle means to find the length of all sides and the measure of all angles. To do this
we need to have one of the following four sets of information:
Case 1: ASA or SAA (one side & 2 angles are known) Case 2: SSA (2 sides and opposite angle are known.
Case 3: SAS: (2 sides & the included angle are known) Case 4: SSS (the three sides are known)
In cases 1 and 2 for a triangle with sides a, b, c and opposite angles , , , respectively, use Law of Sines. In
cases 3 and 4, use law of cosines.
Law of sines: sin  sin   sin 
a
b
c
c 2  a 2  b 2  2ab cos 
Law of Cosines:
b 2  a 2  c 2  2ac cos 
a 2  c 2  b 2  2bc cos
Converting from polar to rectangular coordinates: x = r cos ; y = r sin .
Converting from rectangular to polar coordinates: r 
x 2  y 2 ;   tan 1
y
x
Tests for symmetry: A polar equation is symmetric to the [1] polar axis (x-axis) if (r, ) when replaced by (r, ) yields the same equation; [2] line  = /2 (y-axis) if (r, ) when replaced by (r, -) yields the same equation;
[3] pole (origin) if (r, ) when replaced by (-r, ) yields the same equation. Passing these tests proves that the
equation has the given symmetry; however, equations failing these tests may still have one of the above
symmetries.
In the complex plane, the x-axis becomes the real axis (z = x + oi = x) and the y-axis becomes the imaginary axis (z = 0
+ yi = yi). The magnitude or modulus of z z is the distance from the origin to the point (x, y); z 
x 2  y 2 . If z = x
+ yi is multiplied by its conjugate z = x – yi, the product is x2 + y2 and z 
zz An equation z = x + yi in rectangular
form can be converted to polar coordinates z = r cos  + r sini = r(cos + i sin), r > 0 and 0 <  < 2. In polar form,
the angle  is called the argument of z and r is the magnitude of z.
Product of complex numbers: z1 z2  r1r2 cos  1  2   i sin  1   2   .
z1 r1
 cos  1  2   i sin  1  2   .
z2 r2 
De Moivre’s Theorem: If z = r(cos + i sin) is a complex number, then zn = rn(cos(n)+ sin(n), where n > 1 is a
Quotient of complex numbers:
positive integer.
Complex Roots: Let w = r(cos +i sin) be a complex number and let n > 2 be an integer. If w  0, there are n distinct
complex roots of w given by z  n cos  0  2k    i sin  0  2k   , where k = 0, 1, 2,… n-1.
k
 n

n 
 n

n 
Two vectors, v and w, are equal if the have the same magnitude and direction, v = w. Vector
addition obeys the addition properties of real numbers; ie., commutative, associative, inverse, and
identity. For vector v, -v is a vector with the same magnitude but opposite direction; thus, v – w = v +
(-w). With vectors, real numbers are scalars, which have only magnitude. If α is a scalar and v is a
vector, the scalar product αv obeys the multiplicative properties of real numbers; ie., distributive,
associative, and the properties of 0, 1, and -1. The magnitude of vector v is represented by v and has
the following properties: v > 0; v = 0 if and only if v = 0;  v  v ; and v   v . An
algebraic vector v is shown as v = (a, b), where a and b are real numbers (scalars) called the
components of v. If the initial point of v is the origin in a rectangular coordinate system, then v is a
position vector with terminal point a , b . Any vector with initial point P1  (a1 , b1 ) and terminal point
P2  (a2 , b2 ) is equal to the position vector v  x2  x1 , y2  y1 . A vector u for which u = 1 is called
a unit vector. Any vector v  a , b can be written using the unit vectors i  1, 0 , j  0,1 as follows:
v  a , b  a 1, 0  b 0,1  ai  bj ; a and b are the horizontal and vertical components of v,
respectively. Addition, subtraction, scalar product, and magnitude are defined in terms of the
components of a vector. The vector u = v is a unit vector that has the same direction as a vector v and
v
v  v u . A vector can be written in terms of its magnitude and direction as u  cos i  sin j and
v  v cos i  sin j , where α is the angle between v and i.