Question

Find the period of the function: tan5x

Answer 1

Step-by-step explanation:

By the compounded angle formula, for instance,

sin(x+y)=sin(x)cos(y)+cos(x)sin(y)

one can reduce the size of the angle of a trigonometric function. If the angle is an odd multiple of x, then it can be reduced to a polynomial of the trigonometric function of x of the same kind.

For example,

sin(3x)=sin(2x+x)=sin(2x)cos(x)+cos(2x)sin(x)

Meanwhile,

sin(2x)cos(x)=2sin(x)cos2(x)=2sin(x)(1−sin2(x))=2sin(x)−2sin3(x)

cos(2x)sin(x)=(1−2sin2(x))sin(x)=sin(x)−2sin3(x)

Therefore,

sin(3x)=(2sin(x)−2sin3(x))+(sin(x)−2sin3(x))=3sin(x)−4sin3(x)

Now sin(5x) can also be written into a polynomial of sin(x) in a similar manner. Same for tan(5x).