A reference angle is the smallest possible angle between a given angle measurement and the x-axis.

In this case, our angle $\displaystyle 257^{\circ}$ lies in Quadrant III, so the angle is found by the formula $\displaystyle \angle A_r = \angle A - 180^{\circ}$.

$\displaystyle \angle A_r = \angle A - 180^{\circ}$ $\displaystyle =$ $\displaystyle 257^{\circ} - 180^{\circ} = 77^{\circ}$

Thus, the reference angle for $\displaystyle 257^{\circ}$ is $\displaystyle 77^{\circ}$.

A reference angle is the smallest possible angle between a given angle measurement and the x-axis.

In this case, our angle $\displaystyle 125^{\circ}$ lies in Quadrant II, so we can find our reference angle using the formula

$\displaystyle \angle A_r = 180^{\circ}- \angle A$.

$\displaystyle \angle A_r = 180^{\circ}- \angle A$ $\displaystyle =$ $\displaystyle 180^{\circ} - 125^{\circ} = 55^{\circ}$

Thus, the reference angle for $\displaystyle 125^{\circ}$ is $\displaystyle 55^{\circ}$.

 The period of sinΘ is 2Π, so we set the new angle equal to the base period of 2Π and solve for Θ.

So 4Θ = 2Π and Θ = Π/2.

For the function

$\displaystyle sin\, (Ax)$

the period is equal to

$\displaystyle \frac{2\pi}{A}$

in this case

$\displaystyle \frac{2\pi}{\pi}$

which reduces to $\displaystyle 2$.

For the function

$\displaystyle sin\, (Ax)$

the period is equal to

$\displaystyle \frac{2\pi}{A}$

in this case

$\displaystyle \frac{2\pi}{\frac{1}{2}}=2\pi \cdot \frac{2}{1}$

which reduces to $\displaystyle 4\pi$.

To find the period of Sine and Cosine functions you use the formula:
$\displaystyle \frac{2\pi}{\left | b\right |}$ where $\displaystyle b$ comes from $\displaystyle f(t) = a\sin(bt)$. Looking at our formula you see b is 4 so 
$\displaystyle \frac{2\pi}{\left | 4\right |}=\frac{\pi}{2}$

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

$\displaystyle \omega = \frac{2\pi}{\left|b\right|}$ where $\displaystyle b$ comes from the general formula: $\displaystyle c(t)=a\sin(bt)$. We see that for our equation $\displaystyle \left|b\right| = 2$ and so the period is $\displaystyle \pi$ when you reduce the fraction.

To find period, simply remember the following formula:

$\displaystyle \textup{period}=\frac{2\pi}{B}$

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

$\displaystyle \textup{period}=\frac{2\pi}{4}=\frac{\pi}{2}$

The function $\displaystyle y=-9\sin(5\theta-3)-3$ is related to the parent function $\displaystyle y=\sin(\theta)$, which has a domain of $\displaystyle (-\infty.\infty)$.

The value of theta for $\displaystyle y=-9\sin(5\theta-3)-3$ has no restriction and is valid for all real numbers.  

The answer is $\displaystyle (-\infty,\infty)$.

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

$\displaystyle (-\infty,\infty)$