Question

find the value of sin20°​

Answer 1

Answer:

You should be able to find it using the trig identity: [math]\sin(3x)= 3\sin(x) - 4\sin^{3}(x)[/math]

(I assume this is derived from the identity: sin(x+y) = sin(x)cos(y) + cos(x)sin(y), but used twice. To be honest, I just looked it up.)

Now that we know this, make x=20.

[math]\sin(60)= 3\sin(20) - 4\sin^{3}(20)[/math]

Then make two substitutions. [math]\sin(60)=\frac{\sqrt{3}}{2}[/math] and y = sin(20)

[math]\frac{\sqrt{3}}{2} = 3y - 4y^{3}[/math]

And then with some manipulation:

[math]y^{3} - \frac{3}{4}y + \frac{\sqrt{3}}{8} = 0[/math]

You get 3 solutions. One negative (not correct) the other two are approximately .34 and .64.

Which one is it? sin(30) =.5, and because we know the sine function is increasing up to 90 degrees the solution is approximately .34.

So, what is the exact solution? According to Wolfram Alpha:

Step-by-step explanation:

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