THE SIX TRIGNOMETRIC FUNCTIONS

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ESTABLISHING TRIGONOMETRIC IDENTITIES:

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.