To find the period of a sine, cosine, cosecant, or secant funciton use the formula:

 where  comes from the general formula: . We see that for our equation  and so the period is  when you reduce the fraction.

To find period, simply remember the following formula:

where B is the coefficient in front of x. Thus,

The function  is related to the parent function , which has a domain of .

The value of theta for  has no restriction and is valid for all real numbers.  

The answer is .

For both Sine and Cosine, since there are no asymptotes like Tangent and Cotangent functions, the function can take in any value for . Thus the domain is:

The domain of the tangent function does not include any values of x that are odd multiples of π/2 .

The range of the sine function is from [-1, 1].

The period of the tangent function is π, whereas the period for both sine and cosine is 2π.

The range of a sine wave is altered by the coefficient placed in front of the base equation. So, if you have , this means that the highest point on the wave will be at  and the lowest at .  However, if you then begin to shift the equation vertically by adding values, as in, , then you need to account for said shift.  This would make the minimum value to be  and the maximum value to be . For our question, then, it is fine to use . The  for the function parameter only alters the period of the equation, making its waves "thinner."

The range of a sine wave is altered by the coefficient placed in front of the base equation. So, if you have , this means that the highest point on the wave will be at  and the lowest at ; however, if you then begin to shift the equation vertically by adding values, as in, , then you need to account for said shift. This would make the minimum value to be  and the maximum value to be .  

For our question, the range of values covers . This range is accomplished by having either  or  as your coefficient. ( merely flips the equation over the -axis. The range "spread" remains the same.) We need to make the upper value to be  instead of . To do this, you will need to add  to . This requires an upward shift of .  An example of performing a shift like this is:

Among the possible answers, the one that works is:

The  parameter does not matter, as it only alters the frequency of the function.

The range of the sine and cosine functions are the closed interval from the negative amplitude and the positive amplitude. The amplitude is given by the coefficient,  in the following general equation: 
. Thus we see the range is:

The range of a sine or cosine function spans from the negative amplitude to the positive amplitude. The amplitude is  in the general formula: 
Thus we see amplitude of our function is  and so the range is:

Recall that .

Therefore, we are looking for  or .

Now, this has a reference angle of , but it is in the third quadrant. This means that the value will be negative. The value of  is . However, given the quadrant of our angle, it will be .