To solve the expression [tex]\(4 \sin 2x \cos 4x \sin 6x\)[/tex] given [tex]\(x = 15^\circ\)[/tex], we will proceed step-by-step through the calculations.
1. Convert the angle from degrees to radians:
To use trigonometric functions accurately, we need to convert [tex]\(15^\circ\)[/tex] to radians.
[tex]\[ x = 15^\circ \][/tex]
[tex]\[ x_{\text{rad}} = \frac{15 \pi}{180} = \frac{\pi}{12} \approx 0.2618 \, \text{radians} \][/tex]
2. Calculate the individual trigonometric values:
We need to find the following:
[tex]\[\sin(2x)\][/tex]
[tex]\[\cos(4x)\][/tex]
[tex]\[\sin(6x)\][/tex]
Substitute [tex]\(x = 15^\circ\)[/tex] into these expressions:
[tex]\[ 2x = 2 \cdot 15^\circ = 30^\circ \][/tex]
[tex]\[ 4x = 4 \cdot 15^\circ = 60^\circ \][/tex]
[tex]\[ 6x = 6 \cdot 15^\circ = 90^\circ \][/tex]
- Calculate [tex]\(\sin(2x)\)[/tex]:
[tex]\[ \sin(30^\circ) = \frac{1}{2} \approx 0.5 \][/tex]
- Calculate [tex]\(\cos(4x)\)[/tex]:
[tex]\[ \cos(60^\circ) = \frac{1}{2} \approx 0.5 \][/tex]
- Calculate [tex]\(\sin(6x)\)[/tex]:
[tex]\[ \sin(90^\circ) = 1 \][/tex]
3. Substitute these values into the original expression:
Now we substitute [tex]\(\sin(2x)\)[/tex], [tex]\(\cos(4x)\)[/tex], and [tex]\(\sin(6x)\)[/tex] into the expression [tex]\(4 \sin 2x \cos 4x \sin 6x\)[/tex].
[tex]\[
4 \sin 2x \cos 4x \sin 6x = 4 \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) (1)
\][/tex]
4. Simplify the expression:
[tex]\[
4 \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot 1 = 4 \cdot \frac{1}{4} = 1
\][/tex]
Hence, the value of [tex]\(4 \sin 2x \cos 4x \sin 6x\)[/tex] when [tex]\(x = 15^\circ\)[/tex] is [tex]\(1\)[/tex].
