Section 5-I Angles / Right Triangles / Any angle

where the angle must be in radians
 
 
I. Right Triangle Definition of Trig Functions:

(a) 

(d) 

 

 III. Fundamental Trigonometric Identities:

(a) sin 

(d) sec 

(g) tan 

(j) 1 + 
IV. Cofunctions of complimentary angles:

or rads

Cofunctions of complimentary angles are equal

(a) 

(b)

(c)

 

 V. Function Values of Special Right Triangle Angles:
 
30 2
45 1 1
60 2

VI. x-y Coordinates and Angles

Given terminal point (x, y), length of terminal side = r, and angle 


 
 
VII. Sign of Trig. Functions in each Quadrants:
 
Quadrant II



x < 0
y > 0		 Quadrant I



x > 0
y > 0
Quadrant III



x < 0
y < 0		 Quadrant IV



x > 0
y < 0
VIII. Even / Odd Functions:

A. Even Functions:

B. Odd Functions:

Section 5-II Graphs of sine and cosine functions
 
 

I. Standard form of sine and cosine graphs

y = d + a sin (bx-c) and y = d + a cos (bx - c)

Amplitude = | a | , Largest value of y 

 II. Period of Graph

Given y = a sin bx and y = a cos bx

Period of y  0 < b < 1
III. Translation of sine and cosine graphs

    y = a sin (bx-c) and y = a cos (bx - c)

  c is phase shift

  y = a sin B(x-h), h is horizontal shift
 

  c /   gives the fraction of the period by which the graph shifted horizontally

 IV. Vertical Translation

        y = d + a sin(bx-c) and y = d + a cos (bx-c)

        d represents shift in y position: Graphs y = d + sin x
 

 

Section 5-III Graphs of Other Trigonometric Functions
 
 
I. Graphs of Tangent (tan )

        Model: 
        Period: 
        Domain all 
        Vertical asymptotes: 
        Asymptotes: 

       tan plots from 

 II. Graphs of Cotangent (cot )

        Model: 
        Period: 
        Domain all 
        Vertical asymptotes: 
        Asymptotes: 

        cot plots from 

 

III. Graphs of Reciprocal Functions: secant and cosecant

        Model:  and

or 

        Period: 

 Plot suggestions: to plot csc first plot sin

        To plot sec first plot cos
            a. Intercepts of sin and cos are vertical asymptotes of csc and sec
            b. Maximums of sin and cos are local minimums of csc and sec
            c. Minimums of sin and cos are local maximums of csc and sec

 

Section 5-IV Inverse Trigonometric Functions
 
 
I Inverse Function Properties:

 

 II. Domain and Range of Inverse Functions
Function Domain Range
y = arcsin x => sin y
y = arccos x => cos y
y = arctan x => tan y

 
III. Inverse of sin x

           The y = sin x is a value of the ratio of the 

        The inverse function of y = sin x is denoted by y = sin-1 x or y = arcsin x

        So the arcsin is the angle  whose ratio,  is x

 IV. Inverse of cos x

        The y = cos x is a value of the ratio of the 
        The inverse function of y = cos x is denoted 
        by y = cos-1 x or y = arccos x

         So the arccos is the angle  whose ratio,  is x

        Plot y = arccos x Domain [-1,1] Range[]
V. Inverse of tan x

        The y = tan x is a value of the ratio of the 
        The inverse function of y = tan x is denoted by y = tan-1 x or y = arctan x

        So the arctan is the angle  whose ratio,  is x

Plot of y = tan-1x Domain [] Range[]

Section 6 Verifying / Solving Trigonometric Identities

I. Guidelines for Verifying Trigonometric Identities

    1. Work with one side of an equation at a time

    2. Factor and simplify whenever possible

    3. Use fundamental identities

    4. Try converting all terms to sin and cos

    5. Try anything rather than do nothing - insights comes from trying