Trigonometric Functions

Definition Of Trigonometric Functions For A Right Triangle 

     Triangle $ABC$ has a right angle ($90^circ$) at $C$ and sides of length $a,b,c$ . The trigonometric functions of angle  $A$ are defined as follows.

$1. \sin \text{of} A =\sin A =\cfrac{a}{c} =\cfrac{\text{opposite}}{\text{hypotenuse}}$

$2. cosine \text{of} A = \cos A =\cfrac{b}{c} =\cfrac{\text{adjacent}}{\text{hypotenuse}}$

$3. tangent \text{of} A =\tan A =\cfrac{a}{b} =\cfrac{\text{opposite}}{\text{adjacent}}$

$4. cotangent \text{of} A=\cot A=\cfrac{b}{a}=\cfrac{\text{adjacent}}{\text{opposite}}$

$5. secant \text{of} A=\sec \cfrac{c}{b}=\cfrac{hypotenuse}{\text{adjacent}}$

$6. cosecant \text{of} A=\sec A=\cfrac{c}{a}=\cfrac{\text{hypotenuse}}{\text{opposite}}$

Extensions To Angles Which May Be Greater Than $90^\circ$

     Consider an $xy$ coordinate system . A point  $P$ in the $xy$ plane has  coordinates $(x,y)$ where $x$ is considered as positive along $OX$ and negative along $OX'$ while $y$ is positive along $OY$ and negative along $OY'$ . The distance from origin $O$ to point $P$ is positive and denoted by $r=\sqrt{x^2+y^2}$. The angle $A$ described $counterclockwise$ from $OX$ is considered $positive$ . If it is described $clockwise$ from $OX$ it is considered $negative$. We call $X'OX$ and $Y'OY$ the $x$ and $y$ axis respectively .

     The various quadrants are denotes by I,II,III and IV called the first , second, third and fourth quadrants respectively . In Fig 1 for example, angle $A$ is in the second quadrant while in Fig 2 angle $A$ is in the third quadrant .

$1. \sin A= y/r$

$2. \cos A= x/r$

$3. \tan A=y/x$

$4. \cot A=x/y$

$5. \sec A=r/x$

$6. \csc A=r/y$

Relationship Between Degrees And Radians 

     A $radian$ is that angle $\theta$ subtended at center $O$ of a circle by an arc $MN$ equal to the radius $r$.
     Since $2\pi$ radians = $360^\circ$ we have

1. $1$ radian = $180^\circ /\pi = 57.29577\; 95130\; 8232\; \dots ^\circ$

2. $1^\circ =\pi /180$ radians = $0.0175\; 32925\; 19943\; 29576\; 92\dots$ radians

Relationships Among Trigonometric Functions 

$1. \tan A=\cfrac{\sin A}{\cos A}$

$2. \cot A=\cfrac{1}{\tan A}=\cfrac{\cos A}{\sin A}$

$3. \sec A=\cfrac{1}{\cos A}$

$4. \csc A=\cfrac{1}{\sin A}$

$5. \sin ^2A+\cos ^2A=1$

$6 \sec ^2-\tan ^2 A=1$

$7. \csc ^2A-\cot ^2 A=1$

Sum ,Difference And Product Of Hyperbolic Functions 

$1. sinh x+ sinh y= 2sinh\cfrac{1}{2}(x+y)cosh \cfrac{1}{2}(x-y)$

$2. sinh x-sinh y=2cosh\cfrac{1}{2}(x+y)sinh \cfrac{1}{2}(x-y)$

$3. cosh x+cosh y=2cosh\cfrac{1}{2}(x+y)cosh\cfrac{1}{2}(x-y)$

$4. cosh x-cosh y =2sinh\cfrac{1}{2}(x+y)sinh\cfrac{1}{2}(x-y)$

$5. sinh xsinh y =\cfrac{1}{2}\{cosh(x+y)-cosh(x-y)\}$

$6. cosh xcosh y =\cfrac{1}{2}\{cosh (x+y)+cosh (x-y)\}$

$7. sinh xcosh y=\cfrac{1}{2}\{sinh(x+y)+sinh(x-y)\}$

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