Sure, let's solve the expression [tex]\(2 \sin 2A - 8 \cos A \sin^3 A\)[/tex] step by step, given that [tex]\(A = 1^\circ\)[/tex].
### Step 1: Determine the values of the trigonometric functions
#### Calculate [tex]\(\sin A\)[/tex]
Given [tex]\(A = 1^\circ\)[/tex], [tex]\(\sin 1^\circ \approx 0.01745240643728351\)[/tex].
#### Calculate [tex]\(\cos A\)[/tex]
Given [tex]\(A = 1^\circ\)[/tex], [tex]\(\cos 1^\circ \approx 0.9998476951563913\)[/tex].
### Step 2: Calculate other required trigonometric components
#### Calculate [tex]\(\sin 2A\)[/tex]
Using the double-angle identity [tex]\(\sin 2A = 2 \sin A \cos A\)[/tex],
[tex]\[
\sin 2A = 2 \times 0.01745240643728351 \times 0.9998476951563913 \approx 0.03489949670250097.
\][/tex]
#### Calculate [tex]\(\sin^3 A\)[/tex]
[tex]\[
\sin^3 A = (0.01745240643728351)^3 \approx 5.315767226676433 \times 10^{-6}.
\][/tex]
### Step 3: Substitute the values into the given expression
The original expression to evaluate is:
[tex]\[
2 \sin 2A - 8 \cos A \sin^3 A.
\][/tex]
#### Substituting the values,
[tex]\[
2 \sin 2A \approx 2 \times 0.03489949670250097 \approx 0.06979899340500194.
\][/tex]
[tex]\[
8 \cos A \sin^3 A \approx 8 \times 0.9998476951563913 \times 5.315767226676433 \times 10^{-6} \approx 4.252966087662153 \times 10^{-5}.
\][/tex]
#### Now, combine these results:
[tex]\[
2 \sin 2A - 8 \cos A \sin^3 A \approx 0.06979899340500194 - 4.252966087662153 \times 10^{-5} \approx 0.0697564737441253.
\][/tex]
### Final Answer:
So, the value of the expression [tex]\(2 \sin 2A - 8 \cos A \sin^3 A\)[/tex] when [tex]\(A = 1^\circ\)[/tex] is approximately
[tex]\[
0.0697564737441253.
\][/tex]
