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Verify the following trigonometric identity:

[tex]\[ \sin^3 10^\circ + \cos^3 20^\circ = \frac{3}{4} \left( \cos 20^\circ + \sin 10^\circ \right) \][/tex]


Sagot :

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]