$$ 4 \operatorname{tin} heta=-3 $$ and $$ \cos heta $$ is positive, whe a slated ro coleulate the value of ? $$ 1 \quad \sin heta+3 \cot heta $$ $$ 2 \quad 25 \cos ^{2} heta $$

Question

$$ 4 \operatorname{tin} 	heta=-3 $$ and $$ \cos 	heta $$ is positive, whe a slated ro coleulate the value of ? $$ 1 \quad \sin 	heta+3 \cot 	heta $$ $$ 2 \quad 25 \cos ^{2} 	heta $$

UpStudy AI · Accepted Answer

To solve the problem, we start with the given equation:

$$
4 \tan \theta = -3
$$

From this, we can isolate $$\tan \theta$$:

$$
\tan \theta = -\frac{3}{4}
$$

Since $$\cos \theta$$ is positive, we know that $$\theta$$ is in the fourth quadrant (where $$\tan$$ is negative and $$\cos$$ is positive).

Next, we can find $$\sin \theta$$ and $$\cos \theta$$ using the identity:

$$
\tan \theta = \frac{\sin \theta}{\cos \theta}
$$

Let $$\sin \theta = y$$ and $$\cos \theta = x$$. Then we have:

$$
\frac{y}{x} = -\frac{3}{4} \implies y = -\frac{3}{4} x
$$

Using the Pythagorean identity:

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

Substituting $$y$$ in terms of $$x$$:

$$
\left(-\frac{3}{4} x\right)^2 + x^2 = 1
$$

This simplifies to:

$$
\frac{9}{16} x^2 + x^2 = 1
$$

Combining the terms:

$$
\frac{9}{16} x^2 + \frac{16}{16} x^2 = 1
$$

$$
\frac{25}{16} x^2 = 1
$$

Now, solving for $$x^2$$:

$$
x^2 = \frac{16}{25}
$$

Thus, we find:

$$
\cos^2 \theta = \frac{16}{25}
$$

Now we can find $$\sin^2 \theta$$:

$$
\sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25}
$$

Now we can calculate the two expressions:

1. $$ \sin \theta + 3 \cot \theta $$

First, we need to find $$\sin \theta$$ and $$\cot \theta$$:

$$
\sin \theta = -\sqrt{\frac{9}{25}} = -\frac{3}{5}
$$

$$
\cot \theta = \frac{1}{\tan \theta} = -\frac{4}{3}
$$

Now substituting into the expression:

$$
\sin \theta + 3 \cot \theta = -\frac{3}{5} + 3 \left(-\frac{4}{3}\right)
$$

Calculating $$3 \cot \theta$$:

$$
3 \cot \theta = -4
$$

Thus:

$$
\sin \theta + 3 \cot \theta = -\frac{3}{5} - 4 = -\frac{3}{5} - \frac{20}{5} = -\frac{23}{5}
$$

2. $$ 25 \cos^2 \theta $$

Now we calculate:

$$
25 \cos^2 \theta = 25 \cdot \frac{16}{25} = 16
$$

Now we have both results:

1. $$ \sin \theta + 3 \cot \theta = -\frac{23}{5} $$
2. $$ 25 \cos^2 \theta = 16 $$

Thus, the final answers are:

1. $$ -\frac{23}{5} $$
2. $$ 16 $$
The value of $$ \sin \theta + 3 \cot \theta $$ is $$ -\frac{23}{5} $$ and $$ 25 \cos^2 \theta $$ is 16.

\( 4 \operatorname{tin} \theta=-3 \) and \( \cos \theta \) is positive, whe a slated ro coleulate the value of ? \( 1 \quad \sin \theta+3 \cot \theta \) \( 2 \quad 25 \cos ^{2} \theta \)

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