Evaluating the Trigonometric Expression Sin6° sin66°

The problem asks us to find the value of the trigonometric expression $\sin 6^\circ \sin 66^\circ$. To solve this, we can use a trigonometric identity that converts a product of sines into a sum or difference of cosines. The relevant identity is the product-to-sum formula:

$$2 \sin A \sin B = \cos(A-B) - \cos(A+B)$$

From this, we can write:

$$\sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)]$$

In our expression, we have $A = 6^\circ$ and $B = 66^\circ$. Let's substitute these values into the formula:

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} [\cos(6^\circ - 66^\circ) - \cos(6^\circ + 66^\circ)]$$

Now, let's simplify the angles inside the cosine functions:

• $A - B = 6^\circ - 66^\circ = -60^\circ$
• $A + B = 6^\circ + 66^\circ = 72^\circ$

So the expression becomes:

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} [\cos(-60^\circ) - \cos(72^\circ)]$$

We know that $\cos(-x) = \cos x$. Therefore, $\cos(-60^\circ) = \cos 60^\circ$. The expression is now:

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} [\cos 60^\circ - \cos 72^\circ]$$

Next, we need the values of $\cos 60^\circ$ and $\cos 72^\circ$.

• The value of $\cos 60^\circ$ is a standard trigonometric value: $\cos 60^\circ = \frac{1}{2}$.
• The value of $\cos 72^\circ$ can be related to the sine of $18^\circ$, since $72^\circ = 90^\circ - 18^\circ$. Thus, $\cos 72^\circ = \sin 18^\circ$. The value of $\sin 18^\circ$ is $\frac{\sqrt{5}-1}{4}$. So, $\cos 72^\circ = \frac{\sqrt{5}-1}{4}$.

Substitute these values back into the expression:

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} \left[ \frac{1}{2} - \frac{\sqrt{5}-1}{4} \right]$$

To subtract the fractions inside the brackets, we find a common denominator, which is 4:

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} \left[ \frac{1 \cdot 2}{2 \cdot 2} - \frac{\sqrt{5}-1}{4} \right]$$

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} \left[ \frac{2}{4} - \frac{\sqrt{5}-1}{4} \right]$$

Now, combine the fractions inside the brackets:

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} \left[ \frac{2 - (\sqrt{5}-1)}{4} \right]$$

Distribute the negative sign in the numerator:

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} \left[ \frac{2 - \sqrt{5} + 1}{4} \right]$$

Combine the constant terms in the numerator:

$$\sin 6^\circ \sin 66^\circ = \frac{1}{2} \left[ \frac{3 - \sqrt{5}}{4} \right]$$

Finally, multiply the fractions:

$$\sin 6^\circ \sin 66^\circ = \frac{3 - \sqrt{5}}{8}$$

This matches the value provided in one of the options.