# Why does #cos(90 - x) = sin(x)# and #sin(90 - x) = cos(x)#?

##### 2 Answers

#### Answer

#### Answer:

#### Explanation

#### Explanation:

Note that the image below is only for

If you wish you should be able to draw it with

Definition of

#(# side opposite angle#x)//(# hypotenuse#)#

Definition of

#(# side adjacent to angle#(90^@-x))//(# hypotenuse#)#

but

Therefore

#sin(x) = cos(90^@ -x)#

Similarly

#cos(x) = sin(90^@ - x)#

#### Answer

#### Answer:

#### Explanation

#### Explanation:

Describe your changes (optional) 200

These can also be proven using the sine and cosine angle subtraction formulas:

#cos(alpha-beta)=cos(alpha)cos(beta)+sin(alpha)sin(beta)#

#sin(alpha-beta)=sin(alpha)cos(beta)-cos(alpha)sin(beta)#

Applying the former equation to

#cos(90^@-x)=cos(90^@)cos(x)+sin(90^@)sin(x)#

#cos(90^@-x)=0*cos(x)+1*sin(x)#

#cos(90^@-x)=sin(x)#

Applying the latter to

#sin(90^@-x)=sin(90^@)cos(x)-cos(90^@)sin(x)#

#sin(90^@-x)=1*cos(x)-0*sin(x)#

#sin(90^@-x)=cos(x)#

Describe your changes (optional) 200