19. Gisen that $$ \sin heta=\frac{1}{3} $$, valaulute the numerical value of $$ \sin 3 heta $$, WITIOT wing 2 calculator

Question

19. Gisen that $$ \sin 	heta=\frac{1}{3} $$, valaulute the numerical value of $$ \sin 3 	heta $$, WITIOT wing 2 calculator

UpStudy AI · Accepted Answer

We start with the triple-angle identity for sine:
$$
\sin 3\theta = 3\sin \theta - 4\sin^3 \theta.
$$
Given that $$\sin \theta = \frac{1}{3}$$, we substitute it into the equation:

1. Replace $$\sin \theta$$ with $$\frac{1}{3}$$:
   $$
   \sin 3\theta = 3\left(\frac{1}{3}\right) - 4\left(\frac{1}{3}\right)^3.
   $$

2. Simplify the first term:
   $$
   3\left(\frac{1}{3}\right) = 1.
   $$

3. Compute the cube in the second term:
   $$
   \left(\frac{1}{3}\right)^3 = \frac{1}{27}.
   $$

4. Multiply by $$-4$$:
   $$
   -4\left(\frac{1}{27}\right) = -\frac{4}{27}.
   $$

5. Combine the results:
   $$
   \sin 3\theta = 1 - \frac{4}{27} = \frac{27}{27} - \frac{4}{27} = \frac{23}{27}.
   $$

Thus, the numerical value of $$\sin 3\theta$$ is:
$$
\frac{23}{27}.
$$
The numerical value of $$ \sin 3\theta $$ is $$ \frac{23}{27} $$.

19. Gisen that \( \sin \theta=\frac{1}{3} \), valaulute the numerical value of \( \sin 3 \theta \), WITIOT wing 2 calculator

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