4. Simplify 4.1 $$ \frac{\sin 120 \cdot \cos 210 \cdot an 315 \cdot \cos 27}{\sin 63 \cdot \cos 540} $$

Question

4. Simplify 4.1 $$ \frac{\sin 120 \cdot \cos 210 \cdot 	an 315 \cdot \cos 27}{\sin 63 \cdot \cos 540} $$

UpStudy AI · Accepted Answer

We start with the expression

$$
\frac{\sin 120^\circ \cdot \cos 210^\circ \cdot \tan 315^\circ \cdot \cos 27^\circ}{\sin 63^\circ \cdot \cos 540^\circ}.
$$

**Step 1. Compute individual trigonometric values:**

1. $$ \sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2}. $$

2. $$ \cos 210^\circ = \cos(180^\circ + 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2}. $$

3. $$ \tan 315^\circ = \tan(360^\circ - 45^\circ) = \tan(-45^\circ) = -1. $$

4. $$ \cos 27^\circ $$ remains as it is.

5. $$ \sin 63^\circ $$ remains as it is, but note that
   $$
   \sin 63^\circ = \cos(90^\circ - 63^\circ) = \cos 27^\circ.
   $$

6. $$ \cos 540^\circ = \cos(540^\circ - 360^\circ) = \cos 180^\circ = -1. $$

**Step 2. Substitute the values into the expression:**

$$
\frac{\left(\frac{\sqrt{3}}{2}\right) \cdot \left(-\frac{\sqrt{3}}{2}\right) \cdot (-1) \cdot \cos 27^\circ}{\sin 63^\circ \cdot (-1)}.
$$

**Step 3. Simplify the numerator:**

Multiply the first two factors:

$$
\frac{\sqrt{3}}{2} \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{3}{4}.
$$

Then multiply by $$-1$$:

$$
-\frac{3}{4} \cdot (-1) = \frac{3}{4}.
$$

Thus the numerator becomes:

$$
\frac{3}{4} \cos 27^\circ.
$$

**Step 4. Write the complete expression:**

$$
\frac{\frac{3}{4} \cos 27^\circ}{-\sin 63^\circ}.
$$

Since $$\sin 63^\circ = \cos 27^\circ$$, substitute to get:

$$
\frac{\frac{3}{4} \cos 27^\circ}{-\cos 27^\circ} = \frac{3}{4} \cdot \frac{\cos 27^\circ}{-\cos 27^\circ} = -\frac{3}{4}.
$$

**Final Answer:**

$$
-\frac{3}{4}
$$
$$-\frac{3}{4}$$

4. Simplify 4.1 \( \frac{\sin 120 \cdot \cos 210 \cdot \tan 315 \cdot \cos 27}{\sin 63 \cdot \cos 540} \)