To verify the given equation [tex]\(\sin^3 10^\circ + \cos^3 20^\circ = \frac{3}{4}(\cos 20^\circ + \sin 10^\circ)\)[/tex], let's go through the steps involving the values and expression calculations.
1. Compute [tex]\(\sin 10^\circ\)[/tex]:
[tex]\[
\sin 10^\circ \approx 0.17364817766693033
\][/tex]
2. Compute [tex]\(\cos 20^\circ\)[/tex]:
[tex]\[
\cos 20^\circ \approx 0.9396926207859084
\][/tex]
3. Compute [tex]\(\sin^3 10^\circ\)[/tex]:
[tex]\[
\sin^3 10^\circ \approx (0.17364817766693033)^3 \approx 0.005232187899470064
\][/tex]
4. Compute [tex]\(\cos^3 20^\circ\)[/tex]:
[tex]\[
\cos^3 20^\circ \approx (0.9396926207859084)^3 \approx 0.8297734109401591
\][/tex]
5. Sum of cubes on the left side:
[tex]\[
\sin^3 10^\circ + \cos^3 20^\circ \approx 0.005232187899470064 + 0.8297734109401591 = 0.8350055988396292
\][/tex]
Therefore, the left side of the equation is:
[tex]\[
\sin^3 10^\circ + \cos^3 20^\circ \approx 0.8350055988396292
\][/tex]
6. Right hand side of the equation:
[tex]\[
\frac{3}{4}(\cos 20^\circ + \sin 10^\circ)
\][/tex]
First, we compute the sum inside the parenthesis:
[tex]\[
\cos 20^\circ + \sin 10^\circ \approx 0.9396926207859084 + 0.17364817766693033 = 1.1133407984528388
\][/tex]
7. Multiplying by [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\frac{3}{4}(1.1133407984528388) = \frac{3 \times 1.1133407984528388}{4} \approx 0.835005598839629
\][/tex]
Therefore, the right side of the equation is:
[tex]\[
\frac{3}{4}(\cos 20^\circ + \sin 10^\circ) \approx 0.835005598839629
\][/tex]
Since both sides of the equation give us the same numerical value, we verify that:
[tex]\[
\sin^3 10^\circ + \cos^3 20^\circ \approx \frac{3}{4}(\cos 20^\circ + \sin 10^\circ)
\][/tex]
Both sides equal approximately [tex]\(0.835005598839629\)[/tex], therefore the given equation holds true:
[tex]\[
\sin^3 10^\circ + \cos^3 20^\circ = \frac{3}{4}(\cos 20^\circ + \sin 10^\circ)
\][/tex]
