Questions: Use a coterminal angle to find the exact value of the following expression. Do not use a calculator csc(-1020°) The coterminal angle is 60°. csc(-1020°) = sqrt(3)/2

Solution Steps

To find the exact value of \(\csc(-1020^\circ)\), we first need to find a coterminal angle between \(0^\circ\) and \(360^\circ\). We do this by adding or subtracting \(360^\circ\) until the angle falls within the desired range. Once we have the coterminal angle, we can use known trigonometric values to find the cosecant.

To find a coterminal angle for \(-1020^\circ\), we add \(360^\circ\) repeatedly until the angle is within the range of \(0^\circ\) to \(360^\circ\).

\[ -1020^\circ + 3 \times 360^\circ = 60^\circ \]

Thus, the coterminal angle is \(60^\circ\).

The cosecant function is the reciprocal of the sine function. Therefore, \(\csc(60^\circ) = \frac{1}{\sin(60^\circ)}\).

The sine of \(60^\circ\) is known to be \(\frac{\sqrt{3}}{2}\). Therefore:

\[ \csc(60^\circ) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

To express this in a more standard form, we rationalize the denominator:

\[ \csc(60^\circ) = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

Final Answer