Answer :

Step-by-step explanation:

To find the values of trigonometric functions for given angles, let's use the periodicity and symmetry properties of trigonometric functions.

1. For \(\sin(-1860^\circ)\):

\(\sin(-1860^\circ) = \sin(-1860^\circ + 360^\circ) = \sin(-1500^\circ) = \sin(-1500^\circ + 360^\circ) = \sin(-1140^\circ)\)

Since the sine function is odd, \(\sin(-1140^\circ) = -\sin(1140^\circ)\).

2. For \(\csc(-480^\circ)\):

\(\csc(-480^\circ) = \csc(-480^\circ + 360^\circ) = \csc(-120^\circ)\)

Since the cosecant function is odd, \(\csc(-120^\circ) = -\csc(120^\circ)\).

3. For \(\tan(-225^\circ)\):

\(\tan(-225^\circ) = \tan(-225^\circ + 180^\circ) = \tan(-45^\circ)\)

Since the tangent function is odd, \(\tan(-45^\circ) = -\tan(45^\circ)\).

4. For \(\cos(960^\circ)\):

\(\cos(960^\circ) = \cos(960^\circ - 360^\circ) = \cos(600^\circ)\)

Now, let's find the values of \(\sin(1140^\circ)\), \(\csc(120^\circ)\), \(\tan(45^\circ)\), and \(\cos(600^\circ)\):

1. \(\sin(1140^\circ)\) can be simplified as \(\sin(1140^\circ) = \sin(1140^\circ - 360^\circ) = \sin(780^\circ)\).

2. \(\csc(120^\circ)\) can be simplified as \(\csc(120^\circ) = \csc(120^\circ - 360^\circ) = \csc(-240^\circ)\).

3. \(\tan(45^\circ) = 1\).

4. \(\cos(600^\circ) = \cos(600^\circ - 360^\circ) = \cos(240^\circ)\).

Now, find the exact values using trigonometric ratios for common angles:

1. \(\sin(780^\circ) = \sin(780^\circ - 720^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}\).

2. \(\csc(-240^\circ) = \csc(-240^\circ + 360^\circ) = \csc(120^\circ) = -\csc(120^\circ) = -\frac{2\sqrt{3}}{3}\).

3. \(\tan(45^\circ) = 1\).

4. \(\cos(240^\circ) = -\frac{1}{2}\).

So, the values are:

- \(\sin(-1860^\circ) = -\frac{\sqrt{3}}{2}\)

- \(\csc(-480^\circ) = -\frac{2\sqrt{3}}{3}\)

- \(\tan(-225^\circ) = -1\)

- \(\cos(960^\circ) = -\frac{1}{2}\)