THE SIX TRIGNOMETRIC FUNCTIONS
Composition of Functions with inverses:
They MUST pass the horizontal line test, to do this, we need to restrict the values of the domain.
The inverse of Sin is defined by y=sin-1 x, only if sin y = x. The domain is [-1,1] and the range is restricted to [-π/2, π/2].
example: The sin-1 (1/2) = π/6 because the sin (π/6) = 1/2
The inverse of Cos is defined by y=cos-1x only if x=cos y. The domain is [-1,1] and the range is restricted to [0,π].
example: The cos-1 (-1) = π because the cos (π) = -1
The inverse of Tan is defined by y=tan-1 x only if x=tan y. The domain is (-∞,∞) and the range is restricted to [-π/2,π/2]
example: The tan-1 (1) = π/4 because the tan (π/4) = 1
The inverse of Sec is defined by y=sec-1 x, only if x=sec y. The domain is [1,∞) and the range is restricted to [0,π].
example: The Sec-1 (2) = π/3 because the sec (π/3) = 2
The inverse of Csc is defined by y=csc-1 x, only if x=csc y. The domain is [1,∞) and the range is restricted to [-π/2,π/2].
example: The Csc-1 (2) = π/6 because the csc (π/6) = 2
The inverse Cot is defined by y=cot-1 x only if x=cot y. The domian is (-∞,∞) and the range is restricted to [0,π].
example: The Cot -1 (1) = π/4 because the cot (π/4) = 1
EVEN or ODD IDENTITIES:
Sin (-θ) = -Sin θ
Odd
Cos (-θ) = Cos θ
Even
Tan (-θ) = -Tan θ
Odd
Csc (-θ) = -Csc θ
Odd
Sec (-θ) = Sec θ
Even
Cot (-θ) = -Cot θ
Odd
COSINE FUNCTION
Domain: all real numbers
Range: [-1,1]
Period: 2π
X-intersections: π/2+Kπ (where K is an integer)
Y-intersections: y=1
SECANT FUNCTION
Domain: all real numbers except odd multiples of π/2 or 90 degrees
Range: (-∞,-1]U[1,∞)
Period: 2π
X-intersections: none
Y-intersections: y=1
Inverse of Cosine
COSECANT FUNCTION
Domain: all real numbers except integer multiples of π or 180 degrees
Range:(-∞,-1]U[1,∞)
Period: 2π
X-intersection: none
Y-intersection: none
Inverse of Sine
THE LAW OF COSINES
Standard form:
a2=b2+c2-2bc cos(A)
b2=a2+c2-2ac cos(B)
c2 = a2 + b2 − 2ab cos(C)
Alternate form:
Cos A= b2+c2-a2/2bc
Cos B= a2+c2-b2/2ac
Cos C= a2+b2-c2/2ab
SINE FUNCTION
Y-intersections: y=0
X-intersections: X=Kπ (where K is an integer)
Period: 2π
Range: [-1,1]
Domain: all real numbers
TANGENT FUNCTION
Domain: all real numbers except odd multiples of π/2 or 90 degrees
Range: all real numbers
Period: π
X-intersections: Kπ (where K is an integer)
Y-intersections: y=0
COTANGENT FUNCTION
Domain: all real numbers except integer multiples of π or 180 degrees
Range: all real numbers
Period: π
X-intersections: π/2+Kπ (where K is an integer)
Y-intersections: none
Inverse of Tangent
TRIGONOMETRIC IDENTITIES
Sin t= y
Cos t= x
Tan t= y/x
Csc t= 1/y
Sec t= 1/x
Cot t= x/y
To find the functions that correspond to the Unit Circle we must plug the coordinate values of (x,y) into the identities.
PYTHAGOREAN THEORY:
To find missing sides of right triangles, we use the theorem formula which is: O2+A2=H2 which is just plugin what you have and solve.
Trigonometric Functions: SOHCAHTOA
Sin θ: opposite/hypotenuse
Cos θ: adjasent/hypotenuse
Tan θ: adjacent/opposite
Csc θ: hypotenuse/opposite
Sec θ: hypotenuse/adjacent
Cot θ: opposite/adjacent
For example: in the unit circle we can find the sin π/6=y=1/2
FUNDAMENTAL TRIGONOMETRIC IDENTITIES:
Quotient Identities
Tan θ = sin θ / cos θ
Cot θ = cos θ / sin θ
Pythagorean Identities
Sin2 θ + Cos2 θ = 1
Sin2 θ = Cos2 θ - 1
Cos2 θ = Sin2 θ - 1
Csc2 θ = Cot2 θ + 1
Cot2 θ = Csc2 θ - 1
Tan2 θ + 1 = Sec2 θ
Tan2 θ = Sec2 θ - 1
Reciprocal Identities:
Csc θ= 1/ sin θ
Sec θ= 1/ cos θ
Cot θ= 1/ tan θ
GRAPHING SINE AND COSINE
to do this we must think of several things such as: the five key points, x and y intersections, and their maximum and minimum points.
Basic characteristics of Sin Functions:
Domain: all real numbers
Range: [-1,1]
Period: 2π
X-int every x=Kπ
Y-int at 0
Odd Function
Origin Symmetry
Five Key points of Sin Functions:
(0,0)
(π/2,1)
(π,0)
(3π/2,-1)
(2π,0)
Five Key Points of Cos Functions:
(0,1)
(π/2,0)
(π,-1)
(3π/2,0)
(2π,1)
Basic characteristics of Cos Functions:
Domain: all real numbers
Range: [-1,1]
Period: 2π
X-int every x=π/2 + Kπ
Y-int at 1
Even Function
Y-axis Symmetry
GRAPHING TANGENT, COTANGENT, SECANT, AND COSECANT
Basic characteristics of Tan Functions:
Domain: all real numbers except π/2+Kπ
Range: all real numbers
Period: π
X-int every x=Kπ
Y-int at 0
Vertical Asymptotes every x=π/2+Kπ
Odd Function
Origin Symmetry
Basic characteristics of Cot Functions:
Domain: all real numbers except Kπ
Range: all real numbers
Period: π
X-int every x=π/2+Kπ
None Y-int
Vertical Asymptotes every x=Kπ
Odd Function
Origin Symmetry
Basic characteristics of Sec Functions
Domain: all real numbers except π/2+Kπ
Range: (-∞,-1]U[1,∞)
Period: 2π
None X-int
Y-int at y=1
Vertical Asymptotes every x=π/2+Kπ
Even Function
Y-axis Symmetry
Basic characteristics of Csc Functions
Domain: all real numbers except Kπ
Range: (-∞,-1]U[1,∞)
Period: 2π
None X-int
None Y-int
Vertical Asymptote every x=Kπ
Odd Function
Origin Symmetry
-Sine functions always mirror the x-axis, while Cos functions always mirror the y-axis.
-to graph Csc anbd Sec, we need to find the reciprocal of the y-coordinates of the graphs of sin and cos.
-Sin's maximum corresponds to cosecant's minimum
Verifying Trigonometric Identities:
1. Work at one side at a time
2. Look for opportunities to factor/ add fractions/ square binomials/ combine like terms.
3. Look for opportunities to use the fundamental identities.
4. If all of these fail, then convert everything into sines and cosines.
IMPORTANT: the goal of verifying identities is to establish new identities by manipulating any side of the expression. We cannot multiply /divide or apply any type of condition to the expression, unless it is one of the following methods: Quotient Identities, Reciprocal Identities, Pythagorean Identities, or Even-Odd Identities.
SUM AND DIFFERENCE FORMULAS
For the Cosine function:
cos (a+b)= cos a cos b - sin a sin b
cos(a-b)= cos a cos b + sin a sin b
For the Sine function:
sin(a+b)= sin a cos b + cos a sin b
sin(a-b)= sin a cos b- cos a sin b
For the Tangent function:
tan(a+b)= tan a + tan b / 1 - tan a tan b
tan (a-b)= tan a - tan b / 1 + tan a tan b
DOBLE-ANGLE FORMULAS
sin (2θ) = 2sin θ cos θ
cos (2θ) = cos2 θ - sin2 θ
cos (2θ) = 2cos2 θ - 1
cos2 θ = 1 + cos(2θ) / 2
cos (2θ) = 1 - 2sin2 θ
sin2 θ = 1 - cos(2θ) / 2
tan (2θ) = 2tan θ / 1 - tan2 θ
tan2 θ = 1- cos(2θ) / 1 + cos(2θ)
HALF- ANGLE FORMULAS
sin a/2 = +-√1-cos a / 2
cos a/2 = +- √1+ cos a / 2
tan a/2 = +- √1-cos a / 1+ cos a
tan a/2 = 1 - cos a / sin a = sin a / 1 + cos a
PRODUCT TO SUM FORMULAS
sin a sin b = 1/2 [cos (a-b) - cos (a+b) ]
cos a cos b = 1/2 [cos (a-b) - cos (a+b) ]
sin a cos b = 1/2 [ sin (a+b) + sin (a-b) ]
SUM TO PRODUCT FORMULAS
sin a + sin b = 2 sin a+b/2 cos a-b/2
sin a - sin b = 2 sin a-b/2 cos a+b/2
cos a + cos b = 2 cos a+b/2 cos a-b/2
cos a - cos b = -2 sin a+b/2 sin a-b/2
Sin θ = y/r
Cos θ = x/r
Tan θ = y/x OR Tan θ = sin θ / cos θ
Cot θ = 1 / tan θ OR Cot θ = cos θ / sin θ
Csc θ = 1 / sin θ OR Csc θ = r/y
Sec θ = 1 / cos θ OR Sec θ = 1/x
Why are they identities? Because no matter what value you use in your input, they are always going to be equal.
An equation that is not an identity is called a conditional equations because it has conditions/ restrictions.
If the function is even, then f(-θ) = f(θ) If the function is odd, then f(-θ)= -f(θ
TRANSFORMATIONS OF FUNCTIONS IN GRAPHS
Vertical Shifts
y=f(x) +k the graph is moved up by k units
y=f(x) - k the graph is moved down by k units
Horizontal Shifts
y=f(x+h) the graph is shifted to the left by h units
y=f(x-h) the graph is shifted to the right by h units
Compressing or shifting
y=af(x) you multiply each coordinate by a.
the graph is stretched vertically if a>1
the graph is compressed vertically if 0<a<1
y=f(ax) you multiply each coodinate by 1/a
the graph is stretched horizontally if 0<a<1
the graph is compressed horizontally if a>1
Reflexion
the graph is reflected over the x-axis if y= -f(x)
the graph is reflected over the y-axis if y=f(-x)
THEOREM: The amplitude and period of the graphs are determined by y=Acos(wθ) where |A|= amplitud and T= 2π/w is the period
THE UNIT CIRCLE
A function is called periodic if there is a positive number such as Z where f(θ+Z)= f(θ) this means there are infinite solutions for each equation. The unit circle demonstrates the periodicity of trigonometric functions by showing that they result in a set of values that are repeated in determined intervals.
The sign of a function can be determined by the quadrant of θ that it is landing.
In quadrant I: all functions are positive
In quadrant II: sin θ, csc θ are positive, and the rest of the functions are negative.
In quadrant III: tan θ, cot θ are positive, and the rest of the functions are negative.
In quadrant IV: cos θ, sec θ are positive, and the rest of the functions are negative.