# PreCalc Chapter 5

Ch 5 The Trigonometric Functions

Ch 5.1 Angles

Introducing Positive Angle , Negative Angle , Quadrantal Angle , Coterminal Angle , Acute Angle , Obtuse Angle , Complementary Angle and Supplementary Angle ;
Converting 25°43’37” to Degrees ;
Converting 25.7825 to Degrees, Minutes, Seconds ;

Number of Radian in One Circle ;
Determining Number of Radians of 180°, 90°, 270° in One Revolution of a Circle ;
Definitions of π Radians, π/180 , and 180/π ;
Converting 7π/4 , π/3 to Degrees ;
Converting 3 to Degree, Minutes, Seconds ;
Finding the Arc Length of the AB ;
Finding the Area of the Sector ABθ ;
Definition of Angular Speed and Linear Speed

Ch 5.2 Trigonometric Functions of Angles

SOH, CAH,TOA working with the Acute Angles of a Right Triangle ;
sin(θ)=1/csc(θ), cos(θ)=1/sec(θ) , tan(θ)=1/cot(θ);
csc(θ)=1/sin(θ), sec(θ)=1/cos(θ) , cot(θ)=1/tan(θ);
Finding the 6 Exact Trigonometric Functions if cos(θ)=5/12 ;
Finding the 6 Exact Trigonometric Functions with Legs of 1 ;
Finding the 6 Exact Trigonometric Functions with a Hypotenuse of 2 of a 30-60-90 Triangle ;
Memorizing the Sine, Cosine and Tangent of 30°,45° and 60°;
Memorizing all Three Forms of Pythagorean Identity

Definition of Trigonometric Functions of Any Angle :  r=√x²+y²,  sin(θ)=y/r , cos(θ)=x/r,
tan(θ)=y/x if x≠0 , csc(θ)=r/y  if y≠0 ,  sec(θ)=r/x  if x≠0 , cot(θ)=x/y  if y≠0 ;
The Domain of sin(θ) and cos(θ) ;
The Domain of tan(θ) and sec(θ) ;
The Domain of csc(θ) and cot(θ) ;
Finding Hypotenuse and Adjacent if Opposite=3 and One of the Angles is 60° ;
Finding the 6 Exact Trigonometric Functions if cos(θ)=8/17 ;
Approximating  cos(θ)=(77°), sec(θ)=(202°) , cot(θ)=(-81°) using Calculator

Simplifying 3 csc²(α)-3cot²(α) ;
Simplifying csc(θ)+1 / 1/sin²(θ)+csc(θ) ;
Rewriting tan(θ) in Terms of csc(θ);
Showing  cot(θ)+tan(θ)=csc(θ)sec(θ);
Showing [tan(θ)+cot(θ)]tan(θ)=sec²(θ);
Finding the 6 Exact Trigonometric Functions if cos(θ)=1/2 and sin(θ)<0

Finding the Exact Value of the following Trigonometric Functions without the Use of Calculator
tan(3π/4);
csc(7π/6);
cos(4π/3);
sin(π/2)

Simplifying Trigonometric Expression sin(θ)+cos(θ)cot(θ)

Ch 5.3 Trigonometric Functions of Real Numbers

Trigonometric Functions of on the Unit Circle ;
Unit Circle to Sine Wave and Graphing  y=sin(x);
Finding the Domain, Range,y -intercept, Even or Odd Function, Vertical Asymptote, x-intercept, Period, and Symmetry of  y=sin(x);
Unit Circle to Cosine Wave and Graphing y=cos(x) ;
Finding the Domain, Range, y-intercept, Even or Odd Function, Vertical Asymptote, x-intercept, Period, and Symmetry of  y=cos(x);
Unit Circle to Cosecant Wave and Graphing y=csc(x) ;
Finding the Domain, Range, y-intercept, Even or Odd Function, Vertical Asymptote, x-intercept, Period, and Symmetry of y=csc(x)

Unit Circle to Secant Wave and Graphing y=sec(x) ;
Finding the Domain, Range, y-intercept, Even or Odd Function, Vertical Asymptote, x-intercept, Period, and Symmetry of  y=sec(x);
Unit Circle to Tangent Wave and Graphing y=tan(x);
Finding the Domain, Range, y-intercept, Even or Odd Function, Vertical Asymptote, x-intercept, Period, and Symmetry of  y=tan(x);
Unit Circle to Cotangent Wave and Graphing y=cot(x);
Finding the Domain, Range, y-intercept, Even or Odd Function, Vertical Asymptote, x-intercept, Period, and Symmetry of y=cot(x)

Negative Angle Identities ;
Showing  cot(-x)cos(-x)+sin(-x)=-csc(-x);
As x→0+, then cos(x)→1;
As x→(-π/3) , thencos(x)→1/2 ;
As x→0+, then tan(x)→0;

As x→(-π/2) , then tan(x)→+∞ ;
As x→0+, then sec(x)→1;
As x→(π/2)+ , then sec(x)→-∞

Ch 5.4 Values of Trigonometric Functions

Definition of Reference Angle ;
Finding the Reference Angle for 160°,  -110°,  275°, 4.5 ;
Finding the Reference Angle for cos(-60°) , cos(5π/4) , cot(-150°) , csc(-2π/3) , and their Exact Values , and Determine if the Values are Positive or Negative

Approximating sec(67°50′) and csc(0.26) ;
Solving sin(θ)=0.8225 over [0°, 360°] ;
Solving cos(θ)=-0.6604 over [0°,360°] ;
Solving cot(θ)=1.3752 over [0°, 360°] ;
Solving sec(θ)=-3.51 over [0,2π];
Pythagorean Identities  sin²(θ)+cos²(θ)=1, 1+tan²(θ)=sec²(θ),1+cot²(θ)=csc²(θ)   ;
Definition of Sine, Cosine, and Tangent on the Unit Circle

Ch 5.5 Trigonometric Graphs

Review: How do the Graphs of  y=(x-2)², y=(x+2)², y=x²+2, y=x²-2,  y=2x²,
y=1/2 x²,  y=(2x)², and y=(1/2 x)² Differ from f(x)=x²;
How do the Graphs of  y=2sin(x), y=1/4 sin(x), y=sin(1/4 x), y=sin(3x), y=sin(x)+3, y=sin(x)-1;  y=sin(x-4),y=sin(x+2)    Differ from y=sin(x) ?
Interpreting Trigonometric Function Transformation y=a sin[b(x-c)]+d and y= a cos[b(x-c)+d;
How does the Graph of y= sin[2 (x+π/2)] Differ from y=sin(x) ?

Graphing y=3 sin[2(x- π/4)]+2;
Graphing y= -4 cos(2x+ π/3)

Graphing f(x)=3cos[2(x- π/2)]-1

Interpreting the Sine Graph to Find out Amplitude, Period, Horizontal Shift and Vertical Shift

Trigonometric Function Transformation for f(x)=-2sin(4x-π)+1 and Finding out Amplitude, Period, Horizontal Shift and Vertical Shift

Finding the Equation of the Curve in the Form of y=a sin[b(x-c)]+d and y=a cos[b(x-c)]+d from the Graph with Points (-π/6, 5) and (π,5) ;
Using the Formula f(t)=a cos[b(x-c)]+d to Find the Temperature Function ;
Using the Table to Find the Depth Function of the River

Graphing the Tangent Function y=tan(θ) ;
Using the Unit Circle and the Reciprocal Identity tan(θ)= sin(θ)/cos(θ)

Graphing the Cotangent Function y=cot(θ) using the Graph of y=tan(θ)

Graphing Tangent and over a Different Period y=tan(4x)

Graphing f(x)= -tan(2x)

Determining the Period of  y=tan(3x);
Determining the Period of  y=cot(x/4);
Determining the Amplitude and Period of  y=2 tan(x/2);
Determining the Amplitude and Period of  y=1/2 cot(2x)

How does the Graph of 2cot(πx) Differ from the Graph of y=cot(πx) ?
What are the Zeros of  y=2cot(πx)?

Graphing a Transformation of the Cotangent Function y=2cot(1/4 x)

How would the Graphs y=2tan(x) and y=1/2 tan(x) Differ from the Graph of y=tan(x) ?
Graphing y=1/3 tan(2x-π/4) from y=tan(x);
Graphing y=cot(x+π/4) from y=cot(x);
Graphing y= 1/2 sec(2x-π/2) from y=sec(x);
Graphing y=csc(x+ 3π/4) from y=csc(x);
Graphing  |cos(x)|;
Graphing y=-|sin(x)+2;
Graphing y= x-sin(x) on [-2π,2π] for x and [-6,6] for y ;
Graphing y=|x|cos(x) on [-40,40] for x and [-40,40] for y with y=|x| and y=-|x|