The polar coordinates of a point are given as \(\displaystyle (r,\theta )\), where r represents the distance from the point to the origin, and \(\displaystyle \theta\) represents the angle of rotation. (A negative angle of rotation denotes a clockwise rotation, while a positive angle denotes a counterclockwise rotation.)

The following formulas are used for conversion from polar coordinates to rectangular (x, y) coordinates.

\(\displaystyle x=r\cos\theta\)

\(\displaystyle y=r\sin\theta\)

In this problem, the polar coordinates of the point are  \(\displaystyle \left (2,\frac{-\pi}{4} \right )\), which means that \(\displaystyle r = 2\) and \(\displaystyle \theta=-\frac{\pi}{4}\). We can apply the conversion formulas to find the values of x and y.

\(\displaystyle x=r\cos\theta=2\cos\left (\frac{-\pi}{4} \right )=2\cdot \frac{\sqrt2}{2}= \sqrt{2}\)

\(\displaystyle y=r\sin\theta=2\sin\left (\frac{-\pi}{4} \right )=2\cdot \frac{-\sqrt2}{2}= -\sqrt{2}\)

The rectangular coordinates are \(\displaystyle (\sqrt2,-\sqrt2)\).

The answer is \(\displaystyle (\sqrt2,-\sqrt2)\).

\(\displaystyle \sin X = \frac{OPPOSITE}{HYPOTENUSE}\)

\(\displaystyle \sin C = \frac{AB}{AC}\)

\(\displaystyle \cos\theta\) is defined as the ratio of the adjacent side to the hypotenuse, or in this case \(\displaystyle \frac{y}{x}\). Solving for y gives the correct expression.

The pattern for the side of a \(\displaystyle 30:60:90\) triangle is \(\displaystyle x:x\sqrt{3}:2x\).

 Since \(\displaystyle \cos=\frac{\text{adjacent}}{\text{hypotenuse}}\), we can plug in our given values.

\(\displaystyle \cos=\frac{\text{adjacent}}{\text{hypotenuse}}\)

\(\displaystyle \cos=\frac{x\sqrt{3}}{2x}\)

Notice that the \(\displaystyle x\)'s cancel out.

\(\displaystyle \cos=\frac{\sqrt{3}}{2}\)

Recall that \(\displaystyle sin(\frac{\pi}{6}) =0.5\).

Therefore, we are looking for \(\displaystyle sin(8*\frac{\pi}{6})\) or \(\displaystyle sin(\frac{4\pi}{3})\).

Now, this has a reference angle of \(\displaystyle \frac{\pi}{3}\), but it is in the third quadrant. This means that the value will be negative. The value of \(\displaystyle sin(\frac{\pi}{3})\) is \(\displaystyle \frac{\sqrt{3}}{2}\). However, given the quadrant of our angle, it will be \(\displaystyle -\frac{\sqrt{3}}{2}\).

The problem tells us that the cosine of the angle will be \(\displaystyle \frac{4}{5}\). Cosine is the adjacent over the hypotenuse. From here we can use the Pythaogrean theorem:

\(\displaystyle \text{Opposite}^2+\text{Adjacent}^2=\text{Hypotenuse}^2\)

\(\displaystyle O^2+4^2=5^2\)

\(\displaystyle O^2+16=25\)

\(\displaystyle O^2=9\)

\(\displaystyle \sqrt{O^2}=\sqrt{9}\)

\(\displaystyle O=3\)

Now we know our opposite, adjacent, and hypotenuse.

The cosecant is \(\displaystyle \csc=\frac{\text{Hypotenuse}}{\text{Opposite}}\).

From here we can plug in our given values.

\(\displaystyle \csc=\frac{5}{3}\)

 \(\displaystyle \sec{x}=\frac{\text{hypotenuse}}{\text{adjacent}}\), as it is the inverse of the \(\displaystyle \cos{x}\) function.  This is therefore the answer.

In order to find \(\displaystyle \theta\) we need to utilize the given information in the problem.  We are given the opposite and adjacent sides.  We can then, by definition, find the \(\displaystyle \tan\) of \(\displaystyle \theta\) and its measure in degrees by utilizing the \(\displaystyle \arctan\) function.

\(\displaystyle \tan=\frac{opposite}{adjacent}\)

\(\displaystyle \tan=\frac{10}{8}\)

\(\displaystyle \tan=1.25\)

Now to find the measure of the angle using the \(\displaystyle \arctan\) function.

\(\displaystyle \theta=\arctan1.25\)

\(\displaystyle \rightarrow 51.34^{\circ}\)

If you calculated the angle's measure to be \(\displaystyle 0.90^{\circ}}\) then your calculator was set to radians and needs to be set on degrees.

To get rid of \(\displaystyle \sin(x^\circ)\), we take the \(\displaystyle \arcsin\) or \(\displaystyle \sin^{-1}\) of both sides.

\(\displaystyle \sin(x^\circ)=\frac{1}{2}\)

\(\displaystyle \sin^{-1}(\sin(x^\circ))=\sin^{-1}(\frac{1}{2})\)

\(\displaystyle x^\circ=\sin^{-1}(\frac{1}{2})\)

\(\displaystyle x^\circ=30^\circ\)

In order to find \(\displaystyle \theta\) we need to utilize the given information in the problem.  We are given the opposite and hypotenuse sides.  We can then, by definition, find the \(\displaystyle \sin\) of \(\displaystyle \theta\) and its measure in degrees by utilizing the \(\displaystyle \arcsin\) function.

\(\displaystyle \sin=\frac{opposite}{hypotenuse}\)

\(\displaystyle \sin=\frac{7.6}{15.7}\)

\(\displaystyle \sin=0.4841\)

Now to find the measure of the angle using the \(\displaystyle \arcsin\) function.

\(\displaystyle \theta=\arcsin0.4841\)

\(\displaystyle \rightarrow 28.95^{\circ}\)

If you calculated the angle's measure to be \(\displaystyle \dpi{100} 0.50^{\circ}}\) then your calculator was set to radians and needs to be set on degrees.