Determine whether the functions $$ \mathrm{y}_{1} $$ and $$ \mathrm{y}_{2} $$ are linearly dependent on the interval $$ (0,1) $$. $$ y_{1}=1-2 \sin ^{2} t \cdot y_{2}=6 \cos 2 t $$ Select the correct choice below and, if necessary, fill in the answer box within your choice. A. Since $$ y_{1}= $$ $$ \square $$ $$ y_{2} $$ on $$ (0,1) $$, the functions are linearly independent on $$ (0,1) $$. (Simplify your answer.) B. Since $$ y_{1}=\left(\square y_{2}\right. $$ on $$ (0,1) $$, the functions are linearly dependent on $$ (0,1) $$. (Simplify your answer.) $$ \square $$ (Simplify your answer.) C. Since $$ y_{1} $$ is not a constant multiple of $$ y_{2} $$ on ( 0,1 ). the functions are linearly independent on ( 0,1 ). D. Since $$ y_{1} $$ is not a constant multiple of $$ y_{2} $$ on ( 0,1 ). the functions are linearly dependent on $$ (0,1) $$.

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UpStudy AI · Accepted Answer

We start with the functions

$$
y_{1}=1-2 \sin^2t \quad \text{and} \quad y_{2}=6 \cos2t.
$$

Recall the double-angle identity for cosine:

$$
\cos2t=1-2 \sin^2t.
$$

Thus, we can rewrite $$y_1$$ as

$$
y_{1} = \cos2t.
$$

Now, comparing with $$y_2$$:

$$
y_{2} = 6 \cos2t = 6 y_{1}.
$$

This shows that

$$
y_{1} = \frac{1}{6} y_{2},
$$

which is a constant multiple relationship. Therefore, the functions $$y_1$$ and $$y_2$$ are linearly dependent on the interval $$(0,1)$$.

The correct choice is B with the blank filled in as

$$
\frac{1}{6}.
$$
The functions $$ y_1 $$ and $$ y_2 $$ are linearly dependent on the interval $$ (0,1) $$ because $$ y_1 = \frac{1}{6} y_2 $$.

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