Evaluate the a) $$ an 240^{\circ} $$ b) $$ \sin 300^{\circ} $$

Question

Evaluate the a) $$ 	an 240^{\circ} $$ b) $$ \sin 300^{\circ} $$

UpStudy AI · Accepted Answer

**Step 1. Evaluate $$\tan 240^\circ$$:**

1. Write the angle in terms of an angle in the first rotation:  
   $$
   240^\circ = 180^\circ + 60^\circ.
   $$

2. Use the tangent periodicity property:  
   $$
   \tan(180^\circ + \theta) = \tan \theta.
   $$
   Therefore,  
   $$
   \tan 240^\circ = \tan 60^\circ.
   $$

3. Recall the exact value:  
   $$
   \tan 60^\circ = \sqrt{3}.
   $$

Thus,  
$$
\tan 240^\circ = \sqrt{3}.
$$

---

**Step 2. Evaluate $$\sin 300^\circ$$:**

1. Express the angle in terms of a reference angle:  
   $$
   300^\circ = 360^\circ - 60^\circ.
   $$

2. Use the sine identity for angles in the fourth quadrant:  
   $$
   \sin(360^\circ - \theta) = -\sin \theta.
   $$
   Therefore,  
   $$
   \sin 300^\circ = -\sin 60^\circ.
   $$

3. Recall the exact value:  
   $$
   \sin 60^\circ = \frac{\sqrt{3}}{2}.
   $$

Thus,  
$$
\sin 300^\circ = -\frac{\sqrt{3}}{2}.
$$

---

**Final Answers:**

a) $$\tan 240^\circ = \sqrt{3}$$

b) $$\sin 300^\circ = -\frac{\sqrt{3}}{2}$$
a) $$\tan 240^\circ = \sqrt{3}$$ b) $$\sin 300^\circ = -\frac{\sqrt{3}}{2}$$

Evaluate the a) \( \tan 240^{\circ} \) b) \( \sin 300^{\circ} \)

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