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

$$
\textbf{Step 1: Evaluate each trigonometric function.}
$$

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

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

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

$$
\cos 27^\circ \text{ remains as } \cos 27^\circ.
$$

$$
\text{In the denominator, we have } \sin 63^\circ \text{ and }
$$

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

$$
\textbf{Step 2: Substitute the evaluated values into the expression.}
$$

The original expression is

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

Substituting the values, we get

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

$$
\textbf{Step 3: Simplify the numerator.}
$$

Multiply the first two factors:

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

Including the third factor:

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

Thus, the numerator becomes

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

$$
\textbf{Step 4: Simplify the denominator.}
$$

$$
\sin 63^\circ \cdot (-1) = -\sin 63^\circ.
$$

$$
\textbf{Step 5: Write the combined simplified expression.}
$$

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

$$
\textbf{Step 6: Recognize a trigonometric identity.}
$$

Notice that

$$
\sin 63^\circ = \cos(90^\circ-63^\circ) = \cos 27^\circ.
$$

Substitute this into the expression:

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

$$
\textbf{Final Answer:} \quad -\frac{3}{4}.
$$
The simplified value of the expression is $$-\frac{3}{4}$$.

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