The unit circle is a fundamental concept in trigonometry that helps to understand the periodicity and behavior of trigonometric functions. It shows how functions like sine, cosine, tangent, and their reciprocals (
It is because of this that when solving trigonometric equations, which have no restrictions on theta, we need to find every single possible answer. In representation of all the possible values, we use a formula.
For example: Sin θ = √3/2 has a general formula of θ = π/3 + 2Kπ (where K is an integer.)
MIND MAP
The term ‘mind map’ became popular in the 1970s and was started by Tony Buzan. He was a British psychology author and TV presenter who created this method of brainstorming, also known as ‘radiant thinking.’ Mind mapping helps the mind remember and recall information. It organizes your ideas in a fun way and makes complex thoughts easier to understand. It is flexible and you can erase and add something anytime, meaning you can mix and match multiple ideas in an organized way. I have used mind mapping since the beginning of high school, and it has always been the best way to recall information into my head. Usually, its handwritten and I use a variety of colors and different shapes to know what is most important.
THE UNIT CIRCLE
The sign of a function can be determined by the quadrant of θ that it is landing.
In quadrant IV: cos θ, sec 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 II: sin θ, csc θ are positive, and the rest of the functions are negative.
In quadrant I: all functions are positive
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.
THEOREM: The amplitude and period of the graphs are determined by y=Acos(wθ) where |A|= amplitud and T= 2π/w is the period
TRANSFORMATIONS OF FUNCTIONS IN GRAPHS
Reflexion
the graph is reflected over the y-axis if y=f(-x)
the graph is reflected over the x-axis if y= -f(x)
Compressing or shifting
y=f(ax) you multiply each coodinate by 1/a
the graph is compressed horizontally if a>1
the graph is stretched horizontally if 0
y=af(x) you multiply each coordinate by a.
the graph is compressed vertically if 0
the graph is stretched vertically if a>1
Horizontal Shifts
y=f(x-h) the graph is shifted to the right by h units
y=f(x+h) the graph is shifted to the left by h units
Vertical Shifts
y=f(x) - k the graph is moved down by k units
y=f(x) +k the graph is moved up by k units
If the function is even, then f(-θ) = f(θ) If the function is odd, then f(-θ)= -f(θ
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.
Sec θ = 1 / cos θ OR Sec θ = 1/x
Csc θ = 1 / sin θ OR Csc θ = r/y
Cot θ = 1 / tan θ OR Cot θ = cos θ / sin θ
Tan θ = y/x OR Tan θ = sin θ / cos θ
Cos θ = x/r
Sin θ = y/r
SUM TO PRODUCT FORMULAS
cos a - cos b = -2 sin a+b/2 sin a-b/2
cos a + cos b = 2 cos a+b/2 cos a-b/2
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
PRODUCT TO SUM FORMULAS
sin a cos b = 1/2 [ sin (a+b) + sin (a-b) ]
cos a cos b = 1/2 [cos (a-b) - cos (a+b) ]
sin a sin b = 1/2 [cos (a-b) - cos (a+b) ]
HALF- ANGLE FORMULAS
tan a/2 = +- √1-cos a / 1+ cos a
tan a/2 = 1 - cos a / sin a = sin a / 1 + cos a
cos a/2 = +- √1+ cos a / 2
sin a/2 = +-√1-cos a / 2
DOBLE-ANGLE FORMULAS
tan (2θ) = 2tan θ / 1 - tan2 θ
tan2 θ = 1- cos(2θ) / 1 + cos(2θ)
cos (2θ) = 1 - 2sin2 θ
sin2 θ = 1 - cos(2θ) / 2
cos (2θ) = 2cos2 θ - 1
cos2 θ = 1 + cos(2θ) / 2
cos (2θ) = cos2 θ - sin2 θ
sin (2θ) = 2sin θ cos θ
SUM AND DIFFERENCE FORMULAS
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
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 Cosine function:
cos(a-b)= cos a cos b + sin a sin b
cos (a+b)= cos a cos b - sin a sin b
Verifying Trigonometric Identities:
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.
4. If all of these fail, then convert everything into sines and cosines.
3. Look for opportunities to use the fundamental identities.
2. Look for opportunities to factor/ add fractions/ square binomials/ combine like terms.
1. Work at one side at a time
-Sine functions always mirror the x-axis, while Cos functions always mirror the y-axis.
-Sin's maximum corresponds to cosecant's minimum
-to graph Csc anbd Sec, we need to find the reciprocal of the y-coordinates of the graphs of sin and cos.
GRAPHING TANGENT, COTANGENT, SECANT, AND COSECANT
Basic characteristics of Csc Functions
Vertical Asymptote every x=Kπ
Basic characteristics of Sec Functions
Y-int at y=1
None X-int
Basic characteristics of Cot Functions:
Vertical Asymptotes every x=Kπ
None Y-int
X-int every x=π/2+Kπ
Domain: all real numbers except Kπ
Basic characteristics of Tan Functions:
Vertical Asymptotes every x=π/2+Kπ
Domain: all real numbers except π/2+Kπ
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 Cos Functions:
Y-axis Symmetry
Even Function
Y-int at 1
X-int every x=π/2 + Kπ
Five Key Points of Cos Functions:
(2π,1)
(3π/2,0)
(π,-1)
(π/2,0)
(0,1)
Five Key points of Sin Functions:
(2π,0)
(3π/2,-1)
(π,0)
(π/2,1)
(0,0)
Basic characteristics of Sin Functions:
Origin Symmetry
Odd Function
Y-int at 0
X-int every x=Kπ
FUNDAMENTAL TRIGONOMETRIC IDENTITIES:
Reciprocal Identities:
Cot θ= 1/ tan θ
Sec θ= 1/ cos θ
Csc θ= 1/ sin θ
Pythagorean Identities
Tan2 θ + 1 = Sec2 θ
Tan2 θ = Sec2 θ - 1
Csc2 θ = Cot2 θ + 1
Cot2 θ = Csc2 θ - 1
Sin2 θ + Cos2 θ = 1
Cos2 θ = Sin2 θ - 1
Sin2 θ = Cos2 θ - 1
Quotient Identities
Cot θ = cos θ / sin θ
Tan θ = sin θ / cos θ
For example: in the unit circle we can find the sin π/6=y=1/2
PYTHAGOREAN THEORY:
Trigonometric Functions: SOHCAHTOA
Cot θ: opposite/adjacent
Sec θ: hypotenuse/adjacent
Csc θ: hypotenuse/opposite
Tan θ: adjacent/opposite
Cos θ: adjasent/hypotenuse
Sin θ: opposite/hypotenuse
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.
To find the functions that correspond to the Unit Circle we must plug the coordinate values of (x,y) into the identities.
THE SIX TRIGNOMETRIC FUNCTIONS
TRIGONOMETRIC IDENTITIES
Cot t= x/y
Sec t= 1/x
Csc t= 1/y
Tan t= y/x
Cos t= x
Sin t= y
COTANGENT FUNCTION
Inverse of Tangent
Y-intersections: none
TANGENT FUNCTION
X-intersections: Kπ (where K is an integer)
Period: π
Range: all real numbers
SINE FUNCTION
X-intersections: X=Kπ (where K is an integer)
Y-intersections: y=0
THE LAW OF COSINES
Alternate form:
Cos C= a2+b2-c2/2ab
Cos B= a2+c2-b2/2ac
Cos A= b2+c2-a2/2bc
Standard form:
c2 = a2 + b2 − 2ab cos(C)
b2=a2+c2-2ac cos(B)
a2=b2+c2-2bc cos(A)
COSECANT FUNCTION
Inverse of Sine
Y-intersection: none
X-intersection: none
Range:(-∞,-1]U[1,∞)
Domain: all real numbers except integer multiples of π or 180 degrees
SECANT FUNCTION
Inverse of Cosine
X-intersections: none
Range: (-∞,-1]U[1,∞)
Domain: all real numbers except odd multiples of π/2 or 90 degrees
COSINE FUNCTION
Y-intersections: y=1
X-intersections: π/2+Kπ (where K is an integer)
Period: 2π
Range: [-1,1]
Domain: all real numbers
EVEN or ODD IDENTITIES:
Cot (-θ) = -Cot θ
Sec (-θ) = Sec θ
Csc (-θ) = -Csc θ
Tan (-θ) = -Tan θ
Cos (-θ) = Cos θ
Even
Sin (-θ) = -Sin θ
Odd
Composition of Functions with inverses:
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
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 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 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 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 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
They MUST pass the horizontal line test, to do this, we need to restrict the values of the domain.
