trigonometric identities

Basic Trigonometrics Identities

1. $sin^2 x + cos^x =1$ for all $x \in R$
2. $sec^2x =1 + tan^2x , x \neq (2n+1) \frac {\pi}{2}$
3. $cosec^2 x =1 + cot^2 x, x \neq n \pi$
4. $ sin (2n \pi + x) = sin x$
5. $ sin (n \pi +x) =(-1)^n sin x$
6. $ cos(2n \pi + x) = cos x$
7. $ tan (n \pi + x) = tan x$
8. $ tan (n \pi - x) = -tan x$

Trigonometrics Identities of Sum and difference of angles

Sin and cos function

1. $cos(A+B)=cos(A)cos(B)-sin(A)sin(B)$
2. $cos(A-B)=cos(A)cos(B)+sin(A)sin(B)$
3. $cos(\frac {\pi}{2} -A)=sin(A)$
4. $sin(\frac {\pi}{2} -A)=cos(A)$
5. $sin(A+B)=sin(A)cos(B)+sin(B)cos(A)$
6. $sin(A-B)=sin(A)cos(B)-sin(B)cos(A)$

Similary we can have defined other sin and cos sum and differences

Tan and cot functions

1. If none of the angles x, y and (x + y) is an odd multiple of π/2
   $tan(A+B)=\frac{tan(A)+tan(B)}{1-tan(A)tan(B)}$
   $tan(A-B)=\frac{tan(A)-tan(B)}{1+tan(A)tan(B)}$
   
2. If none of the angles x, y and (x + y) is an multiple of π
   $cot(A+B)=\frac{cot(A)cot(B)-1}{cot(A)+cot(B)}$
   $cot(A-B)=\frac{cot(A)cot(B)+1}{cot(B)-cot(A)}$
   

Some other Important functions

• $cos(A)+cos(B)=2cos\frac{A+B}{2}cos\frac{A-B}{2}$
• $cos(A)-cos(B)=-2sin\frac{A+B}{2}sin\frac{A-B}{2}$
• $sin(A)+sin(B)=2sin\frac{A+B}{2}cos\frac{A-B}{2}$
• $sin(A)-sin(B)=2cos\frac{A+B}{2}sin\frac{A-B}{2}$
  

Trigonometrics Identities of Multiple Angles

Now lets explore the multiple of x. These all can be proved from above equations
Double of x
$cos2x=cos^{^{2}}x-sin^{^{2}}x=2cos^{^{2}}x-1=1-2sin^{^{2}}x=\frac{1-tan^{^{2}}x}{1+tan^{^{2}}x}$
$sin2x=2cos(x)sin(x)=\frac{2tan(x)}{1+tan^{^{2}}x}$
$tan2x=\frac{2tan(x)}{1-tan^{^{2}}x}$
Triple of x
$sin3x=3sin(x)-4sin^{3}x$
$cos3x=4cos^{3}x-3cos(x)$
$tan(3x)=\frac{3tanx-tan^{^{3}}x}{1-3tan^{^{2}}x}$

Conditional Identities

if $A + B + C=180^{\circ}$ and A,B and C are positive angles then
1. $ sin 2A + sin 2B + sin 2C= 4 Sin(A) Sin (B) Sin (C)$
2. $ cos 2A + cos 2B + cos 2C= -1- 4 cos(A) cos (B) cos (C)$
3. $ sin A + sin B + sin C= 4 Sin \frac {A}{2} Sin \frac {B}{2} Sin \frac {C}{2}$
4. for no right angle in A,B,C $ tan A + tan B + tan C=tan(A) tan (B) tan (C)$
5. $ tan(\frac {A}{2}) tan (\frac {B}{2}) +tan(\frac {B}{2}) tan (\frac {C}{2}) + tan(\frac {A}{2}) tan (\frac {C}{2})=1$
6. $ cot A cot B + cot B cot C + cot C cot A=1$

Half Angle Identities

The ones for sine and cosine take the positive or negative square root depending on the quadrant of the angle $\frac {\theta}{2}$. For example, if $\frac {\theta}{2}$ is an acute angle, then the positive root would be used.
1. $sin \frac {\theta}{2}=\pm \sqrt {\frac {1-cos \theta }{2}}$
2. $cos \frac {\theta}{2}=\pm \sqrt {\frac {1+cos \theta }{2}}$
3. $tan \frac {\theta}{2}= \frac { sin \theta}{1 + cos \theta } = \frac { 1- cos \theta}{sin \theta }$
4. $ sin \theta = \frac {2 tan (\theta /2)}{1 + tan^2 (\theta /2)}$
5. $ cos \theta = \frac {1 - tan^2 (\theta /2)}{1 + tan^2 (\theta /2)}$
6. $ tan \theta = \frac {2 tan (\theta /2))}{1 - tan^2 (\theta /2)}$

Trigonometric Ratios for Some Non-common angles

1. $ sin 15^{\circ}=\frac {\sqrt {3} -1}{2 \sqrt {2}} =cos 75^{\circ}$
2. $ cos 15^{\circ}=\frac {\sqrt {3} +1}{2 \sqrt {2}} =sin 75^{\circ}$
3. $ tan 15^{\circ}= \frac {\sqrt {3} -1}{\sqrt {3} +1}$
4. $sin 18^{\circ}=\frac {\sqrt {5} -1}{4}=cos 72^{\circ}$
5. $ tan 18^{\circ}= 2 - \sqrt {3}$
6. $ sin 22 \frac {1}{2}^{\circ}=\frac {\sqrt {2 -\sqrt {2}}}{2}$
7. $ cos 22 \frac {1}{2}^{\circ}=\frac {\sqrt {2 +\sqrt {2}}}{2}$
8. $ tan 22 \frac {1}{2}^{\circ}= \sqrt {2} -1$

Related Topics

Class 11 Maths Class 11 Physics