Trigonometric functions

Trigonometric functions
Definitions of trigonometric functions for a right triangle
A right triangle is a triangle with a right angle (90°)

For every angle ? in the triangle, there is the side of the triangle adjacent to it, the side opposite of it and the hypotenuse such that a^2+b^2=c^2.
For angle ?, the trigonometric functions are defined as follows:
sin??=opp/hyp=( b )/c , cos??=adj/hyp=( a )/c
tan??=sin??/cos?? =opp/adj=( b )/a , cot??=cos??/sin?? =adj/opp=( a )/b
sec??=1/cos?? =hyp/adj=( c )/a , csc??=1/sin?? =hyp/opp=( c )/b
Trigonometric functions of negative angles
sin?(-?)=-sin?? , cos?(-?)=cos?? and tan?(-?)=-tan??
Some useful relationships among trigonometric functions
1. sin^2 x+cos^2 x=1 , sec^2 x-tan^2 x=1 , csc^2?x-cot^2 x=1
2. sin?2x=2 sin?x cos?x , cos?2x=cos^2 x-sin^2 x=1-2sin^2 x=2cos^2 x-1
3. sin^2 x=(1-cos?2x )/2 , cos^2 x=(1+cos?2x)/2



Graphs of Trigonometric Functions


Derivatives of trigonometric functions
If u is a function x, the chain rule version of this differentiation rule is
1.d/dx (sin?u )=cos?u.du/dx
2.d/dx (cos?u )=-sin?u.du/dx
3.d/dx (tan?u )=sec^2 u.du/dx
4.d/dx (cot?u )=-csc^2 u.du/dx
5.d/dx (sec?u )=sec?u tan?u.du/dx
6.d/dx (csc?u )=-csc?u cot?u.du/dx

Example 1: Find derivatives of the functions
1. y=sin^2 x ? y=(sin?x )^2 ? dy/dx=2 sin?x cos?x=sin?2x
2. y=cos?(x^2 ) ? dy/dx=-2x sin?(x^2 )
3. y=tan??x ? dy/dx=sec^2 ?x×1/(2?x)=(sec^2 ?x)/(2?x)
4. y=x^2 sec?3x ? dy/dx=3x^2 sec?3x tan?3x+2x sec?3x=x sec?3x (2+3x tan?3x )
5. y=?(sin?2x ) ? y=(sin?2x )^(1?2)? dy/dx=( 1 )/( 2 ) (sin?2x )^((-1)?2)×cos?2x×2
=( cos?2x )/( ?(sin?2x ) )
Example 2: If y=tan?2t and x=sec?2t show that dy/dx=csc??2t ?
dy/dt=2 sec^2?2t , dx/dt=2 sec?2t tan?2t
dy/dx=dy/dt×dt/dx=2 sec^2?2t×1/(2 sec?2t tan?2t )=sec?2t/tan?2t
=( 1/cos?2t )/(sin?2t/cos?2t )=1/sin?2t =csc??2t ?
Example3: If y=?-cos?? and x=?+cos??;(0????/2)show that dy/dx=(sec??+tan?? )^2
dy/d?=1+sin?? and dx/d?=1-sin??
dy/dx=dy/d?×d?/dx=(1+sin??)/(1-sin?? )
dy/dx=(1+sin??)/(1-sin?? )×(1+sin??)/(1+sin?? )=(1+2 sin??+sin^2??)/(1-sin^2?? )=(1+2 sin??+sin^2??)/cos^2??
dy/dx=1/cos^2?? +(2 sin??)/cos^2?? +sin^2??/cos^2?? =sec^2??+2 sec?? tan??+tan^2??
? dy/dx=(sec??+tan?? )^2
trigonometric functions Inverse
The inverse trigonometric functions are defined to be the inverses of particular parts of the trigonometric functions; parts that do have inverses.
The inverse sine function, denoted by sin^(-1)?x (or arcsin x), is defined to be the inverse of the restricted sine function. A similar idea holds for all the other inverse
trigonometric functions. It is important here to note that in this case the “-1” is not an exponent and so,
sin^(-1)?x?1/sin?x
In inverse trigonometric functions the “-1” looks like an exponent but it isn’t, it is simply a notation that we use to denote the fact that we’re dealing with an inverse trigonometric function. It is a notation that we use in this case to denote inverse trigonometric functions. If we had really wanted exponentiation to denote 1 over sine we would use the following.
(sin?x )^(-1)=1/sin?x

Derivatives of inverse trigonometric functions Let u be a function x, the derivatives of inverse trigonometric functions are:
1.d/dx (sin^(-1)?u )=1/?(1-u^(2 ) ).du/dx 2.d/dx (cos^(-1)?u )=(-1)/?(1-u^(2 ) ).du/dx
3.d/dx (tan^(-1)?u )=1/(1+u^2 ).du/dx 4.d/dx (cot^(-1)?u )=(-1)/(1+u^2 ).du/dx
5. d/dx (sec^(-1)?u )=1/(|u| ?(u^(2 )-1)).(du )/dx 6. d/dx (csc^(-1)?u )=(-1)/(|u| ?(u^(2 )-1)).du/dx
Example 4: Find the derivative for
1.y=sin^(-1)?2x ? dy/dx=1/?(1-(2x)^(2 ) )×2=2/?(1-?4x?^(2 ) )
2.y=3x cos^(-1)?3x-?(1-?9x?^(2 ) )
dy/dx=3x×(-1)/?(1-(3x)^(2 ) )×3+3 cos^(-1)?3x-(-18x)/(2?(1-?9x?^(2 ) ))
=(-9x)/?(1-?9x?^(2 ) )+3 cos^(-1)?3x+9x/?(1-?9x?^(2 ) )=3 cos^(-1)?3x
3.y=2?x tan^(-1)??x
dy/dx=2?x×1/(1+(?x)^2 )×1/(2?x)+2 tan^(-1)??x×1/(2?x)=1/(1+x)+tan^(-1)??x/?x


Find derivative in each of the following problems(1-4)
1. y=sec^2?2x 2. y=x^2 sin?x+2x cos?x-2 sin?x
3. y=?(x^2-1)-sec^(-1)?x 4. y= 2x cos^(-1)??(x )+sin^(-1)??(x )-2?(x-x^2 )
5. If y=1-sin?? and x=?-sin?? find dy/dx
6. If y=sec^(-1)?t and x=?(t^2-1) find dy/dx