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TRIGONOMETRY
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Angles, Arc length, Conversions
Angle measured in standard position.
Initial side is the positive x – axis which is fixed.
Terminal side is the ray in quadrant II, which is free
to rotate about the origin. Counterclockwise rotation
is positive, clockwise rotation is negative.
Coterminal Angles: Angles that have the same terminal side.
60°, 420°, and –300° are all coterminal.
Degrees to radians: Multiply angle by   . 60       radians
180
3
180
Radians to degrees: Multiply angle by
 180 

. 4    45

180
Note: 1 revolution = 360° = 2π radians.
Arc length = central angle x radius, or s  r.
Note: The central angle must be in radian measure.
Right Triangle Trig Definitions
B
c
a
C
•
•
•
•
•
•
b
A
sin(A) = sine of A = opposite / hypotenuse = a/c
cos(A) = cosine of A = adjacent / hypotenuse = b/c
tan(A) = tangent of A = opposite / adjacent = a/b
csc(A) = cosecant of A = hypotenuse / opposite = c/a
sec(A) = secant of A = hypotenuse / adjacent = c/b
cot(A) = cotangent of A = adjacent / opposite = b/a
Special Right Triangles
30°
45°
2
2
3
1
3
cos(30 ) 
2
1
sin( 30  ) 
2
3
tan( 30  ) 
3

1
60°
1
2
3
sin( 60  ) 
2
tan( 60  )  3
cos(60  ) 
1
2
2
2
sin( 45  ) 
2
tan( 45  )  1
cos( 45  ) 
45°
Basic Trigonometric Identities
Quotient identities: tan( A) 
Even/Odd identities:
sin( A)
cos( A)
cos( A)
sin( A)
cot( A) 
cos(  A)  cos( A)
sec(  A)  sec( A)
sin(  A)   sin( A)
csc(  A)   csc( A)
tan(  A)   tan( A)
cot(  A)   cot( A)
Even functions
Odd functions
Odd functions
Reciprocal Identities:
1
csc( A) 
sin( A)
1
sin( A) 
csc( A)
1
sec( A) 
cos( A)
1
cos( A) 
sec( A)
1
cot( A) 
tan( A)
1
tan( A) 
cot( A)
Pythagorean Identities:
sin 2 ( A)  cos 2 ( A)  1
tan 2 ( A)  1  sec 2 ( A)
1  cot 2 ( A)  csc 2 ( A)
All Students Take Calculus.
Quad I
Quad II
Quad III
cos(A)<0
sin(A)>0
tan(A)<0
sec(A)<0
csc(A)>0
cot(A)<0
cos(A)>0
sin(A)>0
tan(A)>0
sec(A)>0
csc(A)>0
cot(A)>0
cos(A)<0
sin(A)<0
tan(A)>0
sec(A)<0
csc(A)<0
cot(A)>0
cos(A)>0
sin(A)<0
tan(A)<0
sec(A)>0
csc(A)<0
cot(A)<0
Quad IV
Reference Angles
Quad I
Quad II
θ’ = 180° – θ
θ’ = θ
θ’ = π – θ
θ’ = θ – 180°
θ’ = θ – π
Quad III
θ’ = 360° – θ
θ’ = 2π – θ
Quad IV
Unit circle
•
•
•
•
•
•
•
Radius of the circle is 1.
x = cos(θ)
 1  cos( )  1
y = sin(θ)
 1  sin(  )  1
2
2
x

y
1
Pythagorean Theorem:
2
2
cos
(

)

sin
( )  1
This gives the identity:
Zeros of sin(θ) are n where n is an integer.

Zeros of cos(θ) are 2  n where n is an
integer.
Graphs of sine & cosine
f ( x)  A sin( Bx  C )  D
g ( x)  A cos( Bx  C )  D
•
•
•
•
•
Fundamental period of sine and cosine is 2π.
Domain of sine and cosine is .
Range of sine and cosine is [–|A|+D, |A|+D].
The amplitude of a sine and cosine graph is |A|.
The vertical shift or average value of sine and
cosine graph is D.
• The period of sine and cosine graph is 2B .
• The phase shift or horizontal shift is CB .
Sine graphs
y = sin(x)
y = 3sin(x)
y = sin(x) + 3
y = sin(3x)
y = sin(x – 3)
y = sin(x/3)
y = 3sin(3x-9)+3
y = sin(x)
Graphs of cosine
y = cos(x)
y = cos(x) + 3
y = 3cos(x)
y = cos(3x)
y = cos(x – 3)
y = cos(x/3)
y = 3cos(3x – 9) + 3
y = cos(x)
Tangent and cotangent graphs
f ( x)  A tan( Bx  C )  D
g ( x)  A cot( Bx  C )  D
• Fundamental period of tangent and cotangent is
π.

• Domain of tangent is x | x  2  n  where n is an
integer.
• Domain of cotangent x | x  n  where n is an
integer.
• Range of tangent and cotangent is .

• The period of tangent or cotangent graph is .
B
Graphs of tangent and cotangent
y = tan(x)
Vertical asymptotes at
x

2
 n .
y = cot(x)
Verrical asymptotes at
x  n .
Graphs of secant and cosecant
y = sec(x)

 n .
Vertical asymptotes at x 
2
Range: (–∞, –1] U [1, ∞)
y = cos(x)
y = csc(x)
Vertical asymptotes at x
Range: (–∞, –1] U [1, ∞)
y = sin(x)
 n .
Inverse Trigonometric Functions
and Trig Equations
y  sin 1 ( x)  arcsin( x)
Domain: [–1, 1]
  
Range:  , 
 2 2
0 < y < 1, solutions in QI and QII.
–1 < y < 0, solutions in QIII and QIV.
1
y  cos ( x)  arccos( x)
Domain: [–1, 1]
Range: [0, π]
0 < y < 1, solutions in QI and QIV.
–1< y < 0, solutions in QII and QIII.
y  tan 1 ( x)  arctan( x)
Domain: 
 
Range:   , 

2 2
0 < y < 1, solutions in QI and QIII.
–1 < y < 0, solutions in QII and QIV.
Trigonometric Identities
Summation & Difference Formulas
sin( A  B)  sin( A) cos( B)  cos( A) sin( B)
cos( A  B)  cos( A) cos( B)  sin( A) sin( B)
tan( A)  tan( B)
tan( A  B) 
1  tan( A) tan( B)
Trigonometric Identities
Double Angle Formulas
sin( 2 A)  2 sin( A) cos( A)
cos( 2 A)  cos 2 ( A)  sin 2 ( A)  1  2 sin 2 ( A)  2 cos 2 ( A)  1
2 tan( A)
tan( 2 A) 
2
1  tan ( A)
Trigonometric Identities
Half Angle Formulas
1  cos( A)
 A
sin    
2
2
1  cos( A)
 A
cos   
2
2
1  cos( A)
 A
tan    
1  cos( A)
2
A
The quadrant of 2
determines the sign.
Law of Sines & Law of Cosines
Law of sines
sin( A) sin( B) sin( C )


a
b
c
a
b
c


sin( A) sin( B) sin( C )
Use when you have a
complete ratio: SSA.
Law of cosines
c 2  a 2  b 2  2ab cos(C )
b 2  a 2  c 2  2ac cos( B)
a 2  b 2  c 2  2bc cos( A)
Use when you have SAS, SSS.
Vectors
• A vector is an object that has a magnitude and a direction.
• Given two points P1: ( x1 , y1 ) and P2: ( x2 , y2 ) on the plane, a
vector v that connects the points from P1 to P2 is
v = ( x2  x1 )i + ( y2  y1 )j.
• Unit vectors are vectors of length 1.
• i is the unit vector in the x direction.
• j is the unit vector in the y direction.
• A unit vector in the direction of v is v/||v||
• A vector v can be represented in component form
by v = vxi + vyj.
2
2
• The magnitude of v is ||v|| = v x  v y
• Using the angle that the vector makes with x-axis in
standard position and the vector’s magnitude, component
form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j
Vector Operations
Scalar multiplication: A vector can be multiplied by any scalar (or number).
Example: Let v = 5i + 4j, k = 7. Then kv = 7(5i + 4j) = 35i + 28j.
Dot Product: Multiplication of two vectors.
Let v = vxi + vyj, w = wxi + wyj.
Example: Let v = 5i + 4j, w = –2i + 3j.
v · w = (5)(–2) + (4)(3) = –10 + 12 = 2.
v · w = vxwx + vywy
Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the
angle between the two vectors.
v θ
w
Two vectors v and w are orthogonal (perpendicular) iff v · w = 0.
Addition/subtraction of vectors: Add/subtract same components.
Example Let v = 5i + 4j, w = –2i + 3j.
v + w = (5i + 4j) + (–2i + 3j) = (5 – 2)i + (4 + 3)j = 3i + 7j.
3v – 2w = 3(5i + 4j) – 2(–2i + 3j) = (15i + 12j) + (4i – 6j) = 19i + 6j.
||3v – 2w|| = 192  62  397  19.9
Acknowledgements
• Unit Circle: http://www.davidhardison.com/math/trig/unit_circle.gif
• Text: Blitzer, Precalculus Essentials, Pearson Publishing, 2006.