To solve the equation [tex]\(4\left(\cos ^3 20^{\circ}+\sin ^3 50^{\circ}\right)=3\left(\cos 20^{\circ}+\sin 50^{\circ}\right)\)[/tex], we will follow several steps to verify the equality. Let's go through each step carefully:
1. Determine [tex]\(\cos 20^\circ\)[/tex] and [tex]\(\sin 50^\circ\)[/tex]:
[tex]\[
\cos 20^\circ \approx 0.9396926207859084
\][/tex]
[tex]\[
\sin 50^\circ \approx 0.766044443118978
\][/tex]
2. Calculate [tex]\(\cos^3 20^\circ\)[/tex] and [tex]\(\sin^3 50^\circ\)[/tex]:
[tex]\[
\cos^3 20^\circ = (0.9396926207859084)^3 \approx 0.8297694655894314
\][/tex]
[tex]\[
\sin^3 50^\circ = (0.766044443118978)^3 \approx 0.4495333323392335
\][/tex]
3. Sum [tex]\(\cos^3 20^\circ\)[/tex] and [tex]\(\sin^3 50^\circ\)[/tex]:
[tex]\[
\cos^3 20^\circ + \sin^3 50^\circ \approx 0.8297694655894314 + 0.4495333323392335 = 1.279302797928665
\][/tex]
4. Multiply the sum by 4:
[tex]\[
4 \left( \cos^3 20^\circ + \sin^3 50^\circ \right) = 4 \times 1.279302797928665 \approx 5.117211191714659
\][/tex]
5. Sum [tex]\(\cos 20^\circ\)[/tex] and [tex]\(\sin 50^\circ\)[/tex]:
[tex]\[
\cos 20^\circ + \sin 50^\circ \approx 0.9396926207859084 + 0.766044443118978 \approx 1.7057370639048865
\][/tex]
6. Multiply the sum by 3:
[tex]\[
3 \left( \cos 20^\circ + \sin 50^\circ \right) = 3 \times 1.7057370639048865 \approx 5.117211191714659
\][/tex]
7. Compare the two sides of the equation:
[tex]\[
4 \left( \cos^3 20^\circ + \sin^3 50^\circ \right) \approx 5.117211191714659
\][/tex]
[tex]\[
3 \left( \cos 20^\circ + \sin 50^\circ \right) \approx 5.117211191714659
\][/tex]
Since both sides of the equation are equal, we can conclude that
[tex]\[
4\left(\cos^3 20^\circ + \sin^3 50^\circ\right) = 3\left(\cos 20^\circ + \sin 50^\circ\right)
\][/tex]
is indeed true. This confirms the given equality.
