Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

If [tex]$x=15^{\circ}$[/tex], then find the value of [tex]$4 \sin 2x \cos 4x \sin 6x$[/tex].

Sagot :

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].