If $$ \sin u=\frac{x}{5} $$, express $$ 3 \sin u+3 \cos u $$ in terms of $$ x $$. Assume $$ 0

Question

UpStudy AI · Accepted Answer

Given that $$ \sin u = \frac{x}{5} $$, we can find $$ \cos u $$ using the Pythagorean identity:

$$
\sin^2 u + \cos^2 u = 1
$$

Substituting $$ \sin u $$:

$$
\left(\frac{x}{5}\right)^2 + \cos^2 u = 1
$$

This simplifies to:

$$
\frac{x^2}{25} + \cos^2 u = 1
$$

Rearranging gives:

$$
\cos^2 u = 1 - \frac{x^2}{25}
$$

To find $$ \cos u $$, we take the square root:

$$
\cos u = \sqrt{1 - \frac{x^2}{25}}
$$

Since $$ 0 < u < \frac{\pi}{2} $$, $$ \cos u $$ is positive.

Now we need to express $$ 3 \sin u + 3 \cos u $$:

$$
3 \sin u + 3 \cos u = 3\left(\sin u + \cos u\right)
$$

Substituting $$ \sin u $$ and $$ \cos u $$:

$$
= 3\left(\frac{x}{5} + \sqrt{1 - \frac{x^2}{25}}\right)
$$

This can be simplified to:

$$
= \frac{3x}{5} + 3\sqrt{1 - \frac{x^2}{25}}
$$

Thus, the expression $$ 3 \sin u + 3 \cos u $$ in terms of $$ x $$ is:

$$
\boxed{\frac{3x}{5} + 3\sqrt{1 - \frac{x^2}{25}}}
$$
$$\frac{3x}{5} + 3\sqrt{1 - \frac{x^2}{25}}$$

If \( \sin u=\frac{x}{5} \), express \( 3 \sin u+3 \cos u \) in terms of \( x \). Assume \( 0

Ask by Cruz Ruiz. in Puerto Rico

Dec 03,2024