Even-odd and periodic properties of trigonometric functions

Question

-sin(3x+2π)+cos(-3x)•tan(π-3x)

Answer 1

 

First the sine function:

 

-sin(3x + 2*pi) = - { sin(3x)*cos(2*pi) + cos(3x)*sin(2*pi)} <---- angle addition for sine

                       = - { sin(3x)* 1           + cos(3x) * 0 }

                       = - sin(3x)

 

This statement is supported by the periodic property which states sin( Z + 2*pi) = sin(Z);

  adding 2*pi does not change the angle at all; it is simply 1 whole revolution around the

  circle and stops at the exact same angle.

 

Next, the cosine function is an ODD FUNCTION,

 so it eats the negative.

cos(-3x) = cos(3x)

 

tan( pi - 3x) =  [tan(pi)- tan(3x)]/[ 1 + tan(pi)*tan(3x) ] <--- angle subtraction formula for tangent

 

                  = [  0 - tan(3x) ]/ [ 1 + 0 * tan(3x) ]   <--- tangent of pi is zero

 

                  = -tan(3x) / 1  = -tan(3x)

 

Substituting these results into the expression:

 

-sin(3x) + cos(3x)*-tan(3x) = 

 

-sin(3x) + cos(3x) * -sin(3x)/cos(3x) =

 

-sin(3x) + -sin(3x) =

 

-2sin(3x)

 

 

To verify and check the answer, I added 2*sin(3x) to the expression

(which is the same as subtracting -2*sin(3x) )

and graphed it in desmos.com

 

Sure enough... flat line... zero.

So the answer is correct.