Expert: Josh - 12/12/2007

Question
Determine the period

1. y = tan (b/2 x)
2. y = tan 2x
3. y = sec
4. if cos x = 1 what is the value sec x?
5. y = sec x
6. y = tan 2x

Answer
Hi Dean

The word "period" derives from the fact that the oscillatory waveform of a sine (or cosine) function repeats itself every so often. If we look at sin(x), you can verify by plugging in numbers that sin(x)=sin(x+360*k) for any integer k. i.e., it repeats itself every 360 degrees. As a result, sin(45)=sin(45+360)=sin(45=720) and so forth. The period is 360.

An angle can be measured in degrees or radians. The two are related by the formula 360 degrees = 2*pi radian, where pi has an approximate value of 3.1415.

e.g., 45 degrees is equivalent to (45/360)*2*pi or pi/4 radian.

All questions will be answered assuming that angles are measured in radians. So, for instance, tan(x)=tan(x+2*pi*k) implies the function has a periodicity of 2*pi (equiv. to 360 deg)

1. How is tan(x) related to tan((b/2)*x)? Changing the argument from x to (b/2)*x simply scales the x-axis. The tangent function is still 2*pi periodic no matter what.
So, tan((b/2)x)=tan((b/2)x+2*pi*k). Next, factorizing everything in terms of the scaling factor "b/2", we see that tan((b/2)x)=tan[(b/2)*(x+2*pi*k/(b/2))]
          =tan[(b/2)*(x+4*pi*k/b)]
Thus, it is 4*pi/b periodic when measured in radians.
[We can readily convert this into degrees, first dividing 4*pi/b by 2*pi, then multiplying it by 360]

Trivial case: If b=2, we have tan((b/2)x)=tan(x). As expected, it is 4*pi/2 or 2*pi periodic.

2. Having 2x instead of x, we are compressing the tangent function by half. Intuitively, the period should reduce by 50%, viz., from 2*pi to pi (or from 360 deg to 180 deg). To verify this, we use the same technique as before.
tan(2x)=tan(2x+2*pi)=tan(2(x+pi)). This indicates a period of pi.

4. sec(x) is the inverse of cos(x). i.e., sec(x)=1/cos(x).
5. Taking the inverse of cos(x) does not alter its periodicity. sec(x)=1/cos(x) remains a 2*pi periodic. function. However, when cos(x)=0, for instance, when x=pi/2 (or 90 deg), sec(x) is undefined.