Trigonometry

Review

The student should read carefully Appendix D in Stewart.

Angles

Definition -- Radians : An angle of radians is the angle that subtends an arc of units when placed at the centre of a unit circle.

==> radians = 180 degrees;

1 radian = degrees; 1 degree = radians

If an angle of radians subtends an arc of s units when placed at the centre of a circle of radius r then or (follows by considering similar figures -- ask me nicely).

It follows that, being a ratio of distances, radians are unitless .

If is obtained by an anticlockwise (counterclockwise) rotation then is positive .

If is obtained by a clockwise rotation then is negative .

>

> convert(1*degrees,radians);evalf(%);

> convert(1,degrees);evalf(%);

Trig Functions

Definition :

If is any angle in standard position and P(x,y) is any point on its terminal side, then:

=

=

= = where = distance from P to the origin.

Note: in Quadrant I , these definitions yield SOHCAHTOA .

The sign of a trig function of depends on which quadrant is in: A ll S udents T ake C alculus.

Definition : If is any Real number then means the sine of radians. Similar definition for each of the trig functions.

Trig functions of special angles. You are expected to know and use these special values. You should memorize the special triangles (see blackboard! ) that they come from rather than memorizing the actual values. You should also know the quadrantal angles and their trig functional values:

Graphs

> restart:

> plot(sin(x),x=-4*Pi..4*Pi,colour=black);

> plot([sin(x),cos(x)],x=-4*Pi..4*Pi,colour=black);

Note : Both the sine and cosine functions are periodic with period .

> plot([sin(x),csc(x)],x=-4*Pi..4*Pi,y=-3..3,colour=black,discont=true);

> plot([cos(x),sec(x)],x=-4*Pi..4*Pi,y=-3..3,colour=black,discont=true);

Note : Both the secant and cosecant functions are periodic with period .

> plot(tan(x),x=-4*Pi..4*Pi,y=-4..4,discont=true,colour=black);

Note : The tangant and cotangent functions are both periodic with period .

> plot(cot(x),x=-4*Pi..4*Pi,y=-4..4,discont=true,colour=black);

Identities

Recall : An identity is a statement about an expression that is true for all values in the domain of the expression.

Even-Odd identities

odd:

even:

Pythagorean identities

note : means [ ][ ] as opposed to etc.

Addition identities

==> =

==>

=

=

Equations

Find all x [0, ] satisfying 2cos(x) - 1 = 0 (#65, pg. A35)

> equ1:=2*cos(x)-1=0;

> solve(equ1,x);

> plot(2*cos(x)-1,x=0..2*Pi,colour=black);

Note: the graph suggests Maple has not given a full answer. FIX!

Find all x [0, ] satisfying (#67, pg. A35)

> plot(2*(sin(x))^2-1,x=0..2*Pi,colour=black);

> solve( 2*sin(x)^2 = 1,x);

Real answer: , , , . FIX!

Find all x [0, ] satisfying sin(2x) = cos(x) (#69, pg. A35)

> equ3:=sin(2*x)=cos(x);

> solve(equ3,x);

> plot(sin(2*x)-cos(x),x=-2*Pi..2*Pi,colour=black);

Note : The plot "says" there are 4 solutions on the given interval. They are: , , , . WHY???!!

Derivatives of Trig Functions

The Big One

> q:=h->(sin(x+h)-sin(x))/h;

> plot({q(1),q(1/2),q(1/5),q(1/10)},x,colour=black);

The plots above suggest that

Let's ask Maple:

> Limit((sin(x+h)-sin(x))/h,h=0)=limit((sin(x+h)-sin(x))/h,h=0);

Theorem : .

The proof of this amazing result depends on knowing four limits:

1) . This seems obvious from "knowing" the graph of y = sin(h). Its proof is not so obvious.

2) This seems obvious and follows easily from the first limit.

3) This is neither obvious nor "easy" to prove.

4) . This is not obvious but follows rather easily from #3.

** You are not expected to be able to prove #1 or #3 but you should be able to use them to prove #2 and #4.

proof : see blackboard.

>

The Other Ones

E.G.s

1. Find f'(x) if .

> exp1:=(sin(x)*tan(x))/x^3;

> exp2:=diff(exp1,x);

> simplify(%);

Working by hand I got exp3 below. Are they the same???

> exp3:=sin(x)*(x*(1+sec(x)^2)-3*tan(x))/x^4;

> exp4:=exp2-exp3;

> simplify(%);

2. Find the equation of the tangent line to at ( , ). (#35, page 149)

> curve:=x*cos(x);

> derivative:=diff(curve,x);

> subs(x=Pi,derivative);

> simplify(%);

> line:=y+Pi=-1*(x-Pi);

> Line:=solve(line,y);

> plot({curve,Line},x,colour=black);